This is Ams Tex file, 78 pages of printed text submitted to JSP BODY \magnification1200 \NoRunningHeads \NoBlackBoxes \def\er{\Bbb R} \def\en{\Bbb N} \def\zet{\Bbb Z} \def\de{\Bbb D} \def\pe{\Bbb P} \def\es{\Bbb S} \def\ex{\Bbb X} \def\ee{\Bbb E} \def\Af{A_{\text{full}}^{*}} \def\Am{A_{\text{max}}^{*} } \def\Gammab{\boldsymbol\Gamma} \def\gb{{\boldsymbol\Gamma}} \def\vv{{V_{\updownarrow}(\gb)}} \def\vvv{{V_{\updownarrow}(\gb')}} \def\vvi{{V_{\updownarrow}(\gb_i)}} \def\vvd{{V_{\updownarrow}(\Cal D)}} \def\vva{{V_{\updownarrow}(A)}} \def\vvb{{V_{\updownarrow}(B)}} \def\vvc{{V_{\updownarrow}(\tilde B)}} \def\vvt{V_{\updownarrow}(T)} \def\vvg{V_{\updownarrow}(G)} \def\ccap{\cap \cap} \def\zv{\Bbb Z^{\nu}} \def\zw{\Bbb Z^{\nu-1}} \def\ps{Pirogov -- Sinai}{} \def\card{|operatorname{card}} \def\diam{\operatorname{diam}} \def\dist{\operatorname{dist}} \def\supp{\operatorname{supp}} \def\card{\operatorname{card}} \def\conn{\operatorname{conn}} \def\con{\operatorname{con}} \def\Conn{\operatorname{Conn}} \def\ssubset{\subset \subset} \def\ext{\operatorname{ext}} \def\loc{\operatorname{loc}} \def\full{\operatorname{full}} \def\sign{\operatorname{sign}} \def\obr#1{\vskip2cm\centerline{\bf}\vskip2cm} \def\df{\flushpar{\bf Definition. }} \topmatter \title Stratified Low Temperature Phases of Stratified Spin Models : A General Pirogov -- Sinai Approach \endtitle \author Petr Holick\'y and Milo\v s Zahradn\'\i{}k \endauthor \affil Faculty of Mathematics and Physics, Charles University, Sokolovsk\'a 83, 186 00 Prague, Czech Republic \endaffil \email holicky\@karlin.mff.cuni.cz, mzahrad\@karlin.mff.cuni.cz \endemail \thanks Partially supported by: Commission of the European Union under contracts CHRX-CT93-0411 and CIPA-CT92-4016, Czech Republic grants 202/96/0731 and 96/272. \endthanks \date August 30, 1997 \enddate \keywords Low temperature Gibbs states, stratified phases (with many interfaces) for stratified and nonstratified Hamiltonians, ``local'' ground states, interfaces, contours, Dobrushin's walls, Pirogov -- Sinai theory, Peierls condition, contour functional, ``metastable'' submodels, (inductively organized) cluster expansions, ground states of one dimensional models, phase diagrams \endkeywords \subjclass 82A25 \endsubjclass \abstract We adapt and improve the existing Pirogov -- Sinai technology to obtain a general and unifying approach to the study of low temperature ``stratified'' phases for classical spin models whose Hamiltonian may not even be translation invariant but is ``stratified'', i.e. invariant with respect to all ``horizontal'' shifts (not changing the last coordinate). Examples are ``stratified'' versions of classical models like the Ising model with ``vertically dependent'' external field; models in halfspaces or layers and also those translation invariant models where Dobrushin's phases with rigid interfaces (one or more) appear. Our method brings some clarification and sharpening even when applied to the ordinary situations of the Pirogov -- Sinai theory. Our main result transcripts the question of characterizing the ``stratified'' Gibbs states of the given model to the question of finding the {\it ground states} of some auxiliary {\it one dimensional\/} model with infinite range but quickly decaying interactions. \endabstract \endtopmatter \document \head I. Introduction, Notes on the Development of the Problem and Some Examples \endhead The rigorous study of Gibbs states with translation noninvariant structure exhibiting a ``rigid interface between the translation invariant phases above and below the interface'', goes back to the pioneering Dobrushin's paper \cite{1}. Several authors continued this study; we note e.g\. article \cite{HKZ} where an attempt to combine basic Dobrushin's ideas with the power of Pirogov -- Sinai theory was made. The leading idea in these investigations was to transcript the problem of description of the structure of the rigid interface to a suitable {\it lower dimensional\/} problem. In more concrete terms, using the expansion of the partition sums above and below the interface, the behaviour of the ``walls'' of the interface between the $+$ and $-$ phases in the three dimensional Ising model can be viewed as a behaviour of contours of some auxiliary two dimensional perturbed Ising model. In \cite{HZ}, we applied a similar approach based on the reduction to a lower dimensional problem to the study of wetting phenomena and entropic repulsion in the Ising model in halfspace. During our attempt to pursue the method to other interesting situations like the study of ``wetting layers'' emerging in some phases of the Blume -- Capel model and also in the order -- disorder -- (other) order phases appering in the Potts model below the critical temperature (the article \cite{HMZ} is under preparation), we found that the additional technical problems are forcing us to look for a more appropriate method. Finally we were lead to a conclusion that the ``dimensional reduction method'' based on this particular kind of a partial exansion of the considered model should be abandoned. Instead, we found a modification of the Pirogov -- Sinai theory which applies {\it directly\/} to these ``stratified'' situations. We hope that the fact that our new version of the Pirogov -- Sinai theory gives even some new insight and a simplification into the traditional ``translation invariant'' Pirogov -- Sinai theory confirms that the method developed by us is adequate. Methodologically, our approach is based on the version \cite{Z} of the Pirogov -- Sinai theory but the concept of a ``stable'' (``small'') contour and of a ``metastable ensemble'' is now investigated in a greater detail. Moreover, the concept of a contour ensemble now {\it disappeared\/} from our version of Pirogov -- Sinai theory completely! The concept of a suitably defined ``contour functional'' $F(\gb)$ (as compared to the contour energy $E(\gb)$) remains in our approach as an extremely important {\it testing quantity\/} -- allowing one to decide whether the contour is ``well behaved enough''-- but instead of the construction of auxiliary contour models, the central point of our approach is now an idea of a successive partial {\it expansion\/} of the model. This is based on an important new technical step which is called {\it recoloring\/} of a contour here. Recoloring of a contour $\gb$ in a partially expanded model means that a new, ``more expanded'' model with the {\it same\/} partition functions is constructed where $\gb$ does not appear anymore. We will see that the ``metastable'' submodels of the given model (constructed for any stratified boundary condition) can be {\it expanded completely\/} and moreover for the ``stable'' boundary conditions, the corresponding metastable model will be identical with the original ``physical'' model. The organization of our expansions will make unnecessary previous estimates like ``Main Lemma'' used in \cite{Z}. Instead, we have now a more powerful method based on our Theorems 5 and 6. To summarize, we converted the Pirogov -- Sinai theory just to a carefully organized {\it method\/} of (successive) {\it expansion of suitable partition functions\/}. The use of expansion techniques is absolutely crucial in our situation and the construction of the expansions is a more delicate task than in the translation invariant situations studied before. Namely, contours of the models studied so far were ``crusted'' in the sense that the events outside and inside the contour were independent. This is {\it not\/} valid here in our new situation, where contours can be often interpreted only as ``walls'' (the terminology of \cite{D}) of the interface(s) and then the events happening ``inside'' resp\. ``outside'' of the wall are {\it not\/} independent ones. This problem was solved in \cite{HKZ} by taking expansions ``above'' and ``below'' the interface and by replacing the walls by more complex ``aggregates'' of walls and clusters \footnote{ The whole situation was then projected to $\zet^{\nu -1}$ which is the main idea of the paper \cite{D}.} but for complicated phases with {\it several\/} interfaces such an approach is too complicated. Now we treat both the ``crusted'' contours and the ``noncrusted'' ones (walls) in the {\it same way\/}. However, the fact that some contours are not ``crusted'' implies that their contour functional $F(\gb)$ (see below for an extensive discussion of this ``testing quantity'') must be now defined much more carefully \footnote{Retaining its meaning, vaguely speaking, of the ``work needed to install the given contour''.}. We construct succesively, by induction, the expansion of the {\it whole metastable model\/}, leaving out the previous idea (of \cite{HKZ}) of the expansion in {\it two\/} different steps (first the expansion of the ensemble of contours and then of the ensemble of the walls resp. of the ``aggregates'', see \cite{HKZ}). \newline Let us now mention some typical examples which can be treated by our method -- which often gives results more powerful than those obtained by previous methods. Besides of the fact that we expect a {\it general applicability\/} of our method to all of these examples, one should stress also that for each special model, the study of the ``important'' terms of the cluster expansion series must be done relevantly to the needs of the particular model to obtain useful results \footnote{For example, to establish the `` logarithmic'' width of the middle layer in the model (2 a,b) below.} and each of the models mentioned below surely requires a detailed treatment. \bigskip (1 a) Models in halfspace $\zv_+=\{t\in\zv;t_{\nu}\geq 0\}$ with ``unstable'' boundary condition on the bottom (like the $-$ boundary condition for the ferromagnetic Ising model with a negative external field, making $+$ the only ground state of the model). Than the ``Basuev states'' (terminology of R\. L\. Dobrushin) with a wetting layer of minuses appear. (1 b) Models in layers (like in \cite{MS}, \cite{MDS}). \smallskip (2 a) Models of Blume -- Capel type with spins belonging to some finite set $Q\subset \er$ and with the Hamiltonian consisting of a (say) quadratic pair interaction and a potential $V$: $$ H(x) =\sum_{(t,s)}(x_t-x_s)^2+\sum_t V(x_t) \tag 1.1 $$ where $V$ has several ``potential wells'' of approximately the same depth. Consider the case of {\it three\/} wells, for example. If $x_+$, $x_0$, $x_-$ mark the bottoms of three adjancent wells of $V$ then it may happen, for suitable choice of $V$, that both $x_+$ and $x_-$ give rise to a stable phase while $x_0$ is unstable. Then one should expect also the existence of a phase which ``goes vertically from $x_+$ to $x_-$ through a layer of a metastable $0$--th phase''. The question is about the determination of the width of the $0$--th layer. On a more concrete level, take the ordinary Blume -- Capel model on $\{-1,0,+1\}^{\zv}$ with a very strong repulsion between $+$ and $-$ and with a weaker, but still strong repulsion between $\pm1$ and $0$ (see \cite{BS} or \cite{ZR} for more information) with the boundary condition $\sign x_t$ being prescribed outside of some finite volume $\Lambda$ and only {\it partially\/}, for $|t_{\nu}| > n $ where $n$ is sufficiently large. The question is what picture is obtained if $\Lambda \to \zv$ , for any extension of the boundary conditions above. Somehow more comfortable variant of such a question is the following one: what happens if we fix the above boundary conditions $+1$ resp. $-1$ for $t_{\nu} \geq n$ resp. $t_{\nu} \leq -n$ and in the middle layer with $|t_{\nu}| \leq n-1$, $ n$ being a suitable integer, we take the spin value zero ? {\it This\/} is the very kind of a question we pose in this paper. The precise value of $\tilde n$ will be known in general only at the very {\it end\/} of our investigation ! Our main theorem says, roughly speaking, that it is possible to prescribe boundary condition for these smaller $|t_{\nu}| $ such that a {\it rigid\/} interface formed by {\it two\/} paralell (slightly perturbed somewhere) planes -- with the layer of (mostly) zero valued spins inside -- appears for {\it any\/} reasonable (finite or infinite) volume $\Lambda$. Of course the subtle question of the exact thickness of such a ``zero phase layer'' inside can be solved only when looking at the quantities $\{h_t\}$ in more detail. This is now a {\it specialized\/} question which can be studied (after the general technology of the computation of these quantities, and {\it this\/} is the essence of the presented paper, was shown to work) for any particular model. Our result assures that the series defining the quantities $\{h_t\}$ very quickly converge and therefore one can give a prescription on precisely how many ``first few terms'' of the expansions have to be computed in order to select, from the competing candidates for the thickness of the layer, the ``best'' one minimizing the free energy of the whole interface. \footnote{ To say this once again: our theorem guarantees the {\it existence\/} of the rigid interface in the infinite volume Gibbs state; however the computation of the precise width of the layer is a more specialized question. A comparison of the competing terms in the expansion series is developed in detail in the prepared article \cite{MZ} -- which deals with the above mentioned Blume -- Capel model and also with the Potts model.} (2 b) A rigid interface with a thick middle layer appears also for the Potts model with a large number of spins slightly below the critical temperature, where phases of the type \newline ``one order above $|$ layer of disorder $|$ another order below'' exist. Using the Fortuin -- Kasteleyn representation this situation can be described analogously as above (see \cite{HMZ}). \smallskip (3 a) ``Sedimentary Ising rock''. ``Stratified \ps\ models''. Consider some ordinary translation invariant Pirogov -- Sinai type model and add to it a small perturbative Hamiltonian which is invariant with respect to the $\Bbb Z^{\nu-1}$ shifts (we identify here $\Bbb Z^{\nu-1}$ with the subspace $\Bbb Z^{\nu-1} \times \{0\}$ of $\zv$) i.e\. depends on the last,``vertical'' coordinate $t_{\nu}$ of $t=(t_1,\dots,t_{\nu})\in \zv$ only. Then, one should expect phases with a possibly rich structure of layers of ``stable or slightly instable translation invariant phases of the unperturbed Hamiltonian''. For example if one adds, to a ferromagnetic Ising model, a small, random (this is the most interesting case) ``horizontally invariant'' external field with approximately zero mean over the vertical shifts, one should expect phases with {\it infinitely many\/} $\pm$ layers and the problem is to compute the positions of the layers. (3 b) In fact, the class of ``horizontally invariant models'' fitting our scheme is much broader and many examples which are not small perturbations of translation invariant models can be constructed. For the illustration of the power of our method consider the situation of the paper \cite{EMZ} (under preparation) where the following situation is studied: take some {\it one dimensional\/} model (either of the Ising type or not; preferably a model with an interesting family of its {\it ground states\/} -- like the translation invariant model producing nonperiodic Thue Morse sequences which is considered in \cite{EMZ}) and consider a {\it three\/} (more generally $\nu$--) dimensional model by adding to the given one dimensional model of ``vertical interactions'' also a sufficiently strong Ising ferrromagnetic ``horizontal interaction'' acting in the directions perpendicular to the $\nu$ -- th axis. Our main theorem assures then that the stratified Gibbs states of such a $\nu$-- dimensional Hamiltonian correspond to the ground states of some small (small for large temperatures) perturbation of the original one dimensional Hamiltonian. In fact in \cite{EMZ} the emphasis is on the {\it opposite\/} question: on the finding of the ``preimage'' of the given interesting one dimensional Hamiltonian, i.e. on the finding of a slightly perturbed Hamiltonian having the property that, when ``stabilized by strong ferromagnetic horizontal interactions'' it yields a family of stratified Gibbs states corresponding exactly to the ground states of the original one dimensional Hamiltonian. Our main result is formulated in detail in part III, section 8, and its summary is given already at the end of part II. Let us outline its meaning: In the translation invariant Pirogov -- Sinai theory, one constructs, for any reference translation invariant configuration (for any translation invariant ``local ground state'') a quantity called the ``metastable free energy''. If the minimum of these quantities is attained for some configuration $y$, then the Gibbs state characterized as the ``local perturbation of $y$'' exists (\cite{Z}). Here, our ``reference configurations'' are (all!) stratified configurations; instead of quantities mentioned above we construct some auxiliary {\it one dimensional\/} model whose configurations correspond to various ``horizontally invariant regimes'' of the original model. The various {\it ground states } of this one dimensional model correspond to the various {\it stratified Gibbs states\/} of our original model. This is our Main Theorem (section 8); the quantities $h_t(y)=h_{t_{\nu}}(y), \ t \in \zv $ constructed there give all the essential information about the model. These quantities are, at least in principle, computable. They are given by cluster expansion series with maybe complicated, but very quickly decaying terms. In the case when $y$ is the ground state of the corresponding one dimensional model (this property will be called the ``stability of $y$'') the quantities $h_{t_{\nu}}(y)$ have the physical interpretation of the ``density of free energy of the $y$--th Gibbs state at the vertical level $t_{\nu}$'' . \remark{Concluding Notes} 1. We are concentrated, in this paper, in the investigation of a phase picture for a {\it fixed\/} Hamiltonian. If one is interested in the investigation of the {\it phase diagram\/} of particular models (notice that there are in principle infinite parameters in the models (3) above!) then such investigation should be based on the study of the mapping: $$ \ \ \{ \ \text{ Hamiltonian} \longmapsto \text{minimizing configurations of }\{h_{t_{\nu}}(y)\} \ \} \tag 1.2$$ Then one can apply theorems from the differential calculus of finitely many (in some models (3) infinitely many) variables, like the implicit function theorem. The study of the {\it differentiability properties\/} of such a diagram requires in fact a suitable technical modification of the definition of the contour functional \footnote{and correspondingly of the quantity $h_t(y) $. Such a modification could however act on the nonground values of $\{h_t(y)\}$ only; the ground values of $\{h_t(y)\}$ have nontrivial physical interpretation and there can be no arbitrariness in their definition!} $F$ -- to obtain sufficiently nice differentiability (even local analyticity) properties of (1.2). 2. We expect that the new method presented in this paper will be applicable also in other situations (even nonstratified ones) where ``noncrusted'' contours appear. Notice also that the method is applicable to situations where one starts (after suitable preparation of the given ``physical'' model, see part II) with some {\it ``mixed model''\/} (see part III, section 2 for the definition of this important concept) instead of the abstract P.S. model. This is (among other examples, like the case of infinite range, quickly decaying potentials we briefly mention below) also the case of models with continuous spins (studied in \cite{DZ}) having several ``potential wells''. In these models, the expansion around (positive mass) gaussians (approximating the regimes of the potential wells) of the ``restricted ensembles'' (of configurations living in the vicinity of the potential wells) yields a mixed model of the type studied here, and then the analysis developed in part III of this paper could be applied to these models, possibly also for the wells which are not so ``deep'' such that the previous analysis (like \cite{DZ}) could be applied to them. \endremark \definition{Acknowledgements} The second author (M.Z.) thanks the Erwin Schr\"odinger Institute for its hospitality during the time of the autumn 1995 semester ``Gibbs random fields and phase transitions''. Unfortunately, the organizer of the semestr and our teacher R.L. Dobrushin could not already come. We dedicate this paper to his memory. \enddefinition \head II. General Description of the Considered Model. Transcription to an Abstract Pirogov -- Sinai Type Model \endhead Given a configuration space $$ \Bbb X= S^{\zv}\,, \,\,\, \nu\geq 3 , \tag 2.1$$ where $S$ is a finite set (of ``spins''), we will consider a general ``horizontally invariant'' (called also ``stratified'' below) Hamiltonian on $X$. This Hamiltonian $H$ will have (at least in the basic case studied below) a finite range $r \in \en$. \footnote{Our method can be immediately extended to some more general {\it infinite range\/} Hamiltonians with sufficiently quickly (exponentially!) decaying interactions. However, here it is not a proper place to formulate such a generalization. For some more details see Corollary, end of part II and also the discussion after Main Theorem, part III. The paper in preparation \cite{EMZ} will give more information about this problem.} In what follows we have to consider a suitable norm on $\zv$ ; let us fix the usage of the $l_{\infty}$ norm $|\cdot\|=|\cdot\|_\infty$ \footnote{ In fact, the choice of the norm $l_1$ would be in a sense nicer, stronger and more natural in some parts of the paper, like in the beginning of part III (if one relates it to our very notion of commensurability). However, in other reasonings the $l_{\infty}$ norm is more convenient. For example, the cubes introduced in the later sections of part III should be otherwise replaced by octagons to get the strongest possible estimates there. We do not want to do that in this paper.} everywhere. Put $$ H(x_{\Lambda }|x_{\Lambda^c })= \sum \Sb A\cap\Lambda \ne\emptyset\\ \diam A \leq r \endSb \Phi _A(x_A) , \tag 2.2 $$ where $\Phi_A$ are some ``interactions'', i.e. functions on $S^{A}$ with values in $\er \cup +\infty$ which are``stratified'' in the sense explained below. We will study the structure of (stratified, see below) {\it Gibbs states\/} of the model, more precisely of the probabilities \footnote{The probabilities $ P_{\Lambda,\beta} ^{x_{\Lambda ^c}} (\cdot)$ are called finite volume Gibbs states under boundary condition $x_{\Lambda ^c}$. } which are given in finite volumes $\Lambda $ by formulas $$ P_{\Lambda,\beta}^{x_{\Lambda ^c}} (x_\Lambda )= (Z_{\beta}(\Lambda ,x_{\Lambda ^c}))^{-1} \exp(-\beta H(x_{\Lambda }|x_{\Lambda ^c})) \tag 2.3 $$ where $\beta =\frac{1}{T}$ is the inverse temperature. The {\it partition function\/} $ Z_{\beta}(\Lambda ,x_{\Lambda ^c})$, under boundary condition $x_{\Lambda^c} \in S^{\Lambda^c}$ is defined by $$ Z_{\beta}(\Lambda, x_{\Lambda ^c})= \sum_{x_{\Lambda }} \exp(-\beta H(x_{\Lambda }|x_{\Lambda ^c})) . \tag 2.4$$ Suitable {\it infinite volume limits\/} of these finite volume Gibbs states will be studied below. \subhead Notes \endsubhead {\bf 1.} In fact, some other (more special than (2.4)) partition functions, namely so called (strictly) {\it diluted\/} partition functions will be important later. The recurrent structure of the measures (2.3) -- which is formulated by the DLR equations -- and the concept of a general (nondiluted) partition function under arbitrary boundary condition will not play, in our later approach, such an important role as usually. More adequate for our later approach is the idea that $\Lambda$ is some (typically very large) volume which will be {\it fixed\/} in the main part of our future considerations, together with some very special (``stratified'', see below) boundary conditions given on its complement. Only at the very end of the paper -- when proving and interpreting our Main Theorem, section III.8 -- our ``playground'' $\Lambda$ will be expanded to the whole $\zv$ and the limit Gibbs states thus obtained will be investigated.\newline {\bf 2.} In the following we will {\it always put\/} $\beta=1$\ i.e. \ we will include the term $\beta = 1/T$ into the definition of $\Phi _{A}$ and $H$. Thus the temperature \footnote{{\it Low\/} temperature in most applications; see the Peierls condition formulated below in (2.14).} will be just one of the parameters in the Hamiltonian. We emphasize that in this paper we are interested only in the clarification of the phase picture for a given {\it fixed\/} Hamiltonian. Doing this, one can study the {\it change\/} of this picture (and of relevant quantities like the free energies) when some parameters are changing. Our approach gives some basic tools for doing that: namely we define useful quantities called {\it metastable free energies\/} which really govern the behaviour of the phase diagram -- see our Main Theorem. However, relevant result on the behaviour of the phase {\it diagram\/} is, in a sense, not even formulated in our paper. (See Corollary at the end of section II for some information.) \newline {\bf 3.} One could be interested in the structure of Gibbs states, under suitable boundary conditions, also for other infinite volumes like the halfspace $\zv_+=\{t\in\zv;t_{\nu}\geq 0\}$ or in a layer. It is not hard to see that such a situation could be modeled on $\Lambda =\zv$ by choosing a suitable modification of the Hamiltonian: for example if we put $$ \Phi _{A}(x_{A})=+\infty $$ whenever $A\not\subset\zv_+$ and $x_{A\cap \zv_{-}}\ne\{x_t=\bar x_{t}, t\in A \cap \zv_{-}\}$ we obtain a limit Gibbs state on $\zv_{+}$ under the boundary condition $\bar x$ on $\zv_{-}=\zv\setminus\zv_+$. Take, for example, $\bar x = +$ for the Ising model with a positive magnetic field. This yields the ``Basuev'' state on $\zv_{+}$. \newline {\bf 4.} In fact, meaningful and nontrivial results requiring the full strength of all the forthcoming constructions can be formulated even without taking the thermodynamic limit, for a fixed {\it finite\/} volume $\Lambda$ (imagine the cardinality $|\Lambda | = 10^{27}$ !), with suitable boundary conditions. However, in such a case it is of course natural to study also a torus with periodic boundary conditions. Though we do not work out here the (topological) modifications needed to carry on our study also to the case of periodic boundary conditions, it seems that only minor parts of the text should be modified or replaced by another arguments (for example, the parts of the text where the lexicographic order is used) to apply our method. \vskip1mm \head 1. Stratified configurations, Hamiltonians and States \endhead For any $u\in\zw$ consider the shift \ $ U \equiv \{t \mapsto t+(u,0)\}\, : \zv\rightarrow\zv $ and correspondingly define the shifts $ \{x \mapsto U(x)\}\, : \ S^{A}\rightarrow S^{U(A)} $ where $U(x)=\tilde x$ has coordinates $\tilde x_{t+u}=x_t$, $t\in A$ and $ \Phi _{A} \mapsto (U\Phi) _{U(A)} $ \ where $(U\Phi) _{U(A)}(U(x_{A})) = \Phi _{A}(x_{A})$. Say that a configuration $x$ is {\it stratified\/} (or {\it horizontally\/} {\it invariant\/}) if $ U(x)=x\quad \text{for each } u\in\zw $ where we identify $\zw$ as a subset $ \zw = \{(t_1,\dots,t_{\nu -1},0)\} \subset \zv\,. $ \definition{Notation} We denote by $\es\subset \ex$ the collection of {\it all stratified configurations\/}.\enddefinition Analogously we define the notion of a {\it stratified Hamiltonian\/} $H=\{ \Phi _{A} \}$ and a {\it stratified\/}\ (Gibbs) {\it measure\/} $\mu $ by requiring, for each $u \in \zw$, $$ \{ (U\Phi)_{A}\}=\{ \Phi _{A}\}\,,\,\,\,U(\mu )=\mu. $$ \remark{ Notes} {\bf 1}. These configurations will be the ``local ground states'' of our model. Of course only {\it some\/} of them will ``deserve'' this name. Analogously, in the traditional Pirogov -- Sinai situation, only some of the constant (resp\. periodic) configurations ``deserve'' the name of ``local ground state''. However, it is often problematic to separate these ``true local ground states'' from the other horizontally invariant (analogously, translation invariant resp\. periodic) configurations. Moreover, the ``reference family'' $\es$ will have to be chosen such that whenever $x \in \es$ locally (i.e. $x$ is equal to some $ y \in \es$ in the $r$ --neighborhood of each point) then we have (``globally'') $x \in \es$. Also, nonexistence of a substantial energetic barrier between the ``true'' local ground states and the remaining elements of $\es$ requires some care in the formulation of the Peierls condition, see below. The simplest solution in such a situation is to choose the reference family $\es$ as big as above, and to look for energetical barriers only between $\es$ and configurations which are {\it not\/} stratified. \footnote{This is an interesting methodological point even for the ordinary Pirogov -- Sinai theory. Namely, we now suggest to consider the family of {\it all\/} translation invariant configurations as the ``reference family''of configurations in the ordinary Pirogov --Sinai setting. Such an approach leads, in fact, to a sharper and clearer formulation of the Peierls condition. (See below in (2.14) for our analogous situation.)} \newline {\bf 2.} The framework when all the local ground states of the model are are chosen among the stratified ones (analogously, compare the choice of translation invariant configurations in the ordinary Pirogov -- Sinai theory) seems at the first sight to be too narrow in the situations (like Ising antiferromagnet) where {\it periodical\/} (local) ground states occur. However, it is easily seen that periodical resp\. horizontally periodical configurations can be converted to constant resp\. stratified ones by taking suitable {\it blockspin transformation\/} (over the periods) and so the setting we introduce here is sufficiently general. \newline {\bf 3.} The fact that we are selecting {\it several\/} configurations (in fact the whole family $\es$) as the ``reference'' ones -- expecting that some of these configurations may (possibly, under suitable adjustment of the Hamiltonian) give rise to corresponding Gibbs states -- suggests that our interest lies in the situations where phase transitions {\it may occur\/}. Thus, the possible ``degeneracy of the ground state'' \footnote{ By the {\it degeneracy\/} of a ground state one usually means that {\it several\/} ground states exist for a given Hamiltonian. Here, we are looking for ground states (resp. more generally for ``local ground states'') only among the elements of $\es$, i.e. we are looking for configurations $y \in \es$ such that $\sum_A (\Phi_A(x_A) -\Phi_A(y_A)) > 0 $ whenever $x$ differs from $y$ on a set whose vertical size is finite (resp. ``not too big'').} is the situation of our interest. Though in most situations we will have to deal with only {\it one\/} Gibbs state corresponding to a given Hamiltonian, we want to have a theory dealing at the same time with the situations of phase coexistence. This requirement distinguishes the \ps\ theory from the methods focused on the study of the {\it phase uniqueness\/} region. Namely, recall that in the region of phase uniqueness other and well developed methods of study (based essentially on the Dobrushin' s unicity theorem and on the later investigation of the ``complete analyticity'' properties by \cite{DSA}) are available. On the other hand, in the regions where phase coexistence is expected apparently no serious alternative to the \ps\ theory exists reaching a comparable level of generality and universality of its applications. \endremark \vskip1mm \head 2. Precontours and Admissible Systems of Precontours \endhead \definition{Stratified points of a configuration $x$} Given a configuration $x\in\ex$ say that a point $t\in\zv$ is a {\it stratified\/} point of $x$, more precisely {\it $y$-stratified\/} (for $y\in\es$) point of $x$, \ if the equality $ x_{\tilde t}=y_{\tilde t} $ holds for each $\tilde t\in\zv$ such that $||\tilde t-t||_{ \infty} \leq r$. \enddefinition \remark{Note}This is an analogy of the notion of a correct point of the ordinary Pirogov -- Sinai theory; $r$ is the range of the interactions in the given Hamiltonian (2.2). \endremark \definition{Diluted configurations, external colour of a configuration} Say that $\Lambda\subset \zv$ is a {\it standard set\/} if all the sets $C_n=\{t\in\zv;\, t_\nu=n\}\setminus\Lambda$ are {\it simply connected\/}, i.e. if any two points $t,t' \in \Lambda ^{c}$, resp. $ \Lambda$, with the same last coordinate $t_{\nu}=t_{\nu}'$ can be connected by a ``horizontal'' (keeping the last coordinate intact) connected path in $\Lambda ^c $, resp. $ \Lambda$. A configuration $x\in\ex$ will be called {\it $y$ -- diluted\/}, for $y \in \es$, if there is some set \ $ \ee\subset\zv $ \ whose all connected components are finite standard sets (an infinite number of components is allowed) and such that all the points of $\ee^c$ are $y$ -- stratified. \footnote{This traditional notion of \ps \ theory is {\it not\/} best suited for our purposes. It will be {\it strenghtened\/} below. See the notion of an {\it isolation (and strict diluteness)\/} below in (2.5) (and (2.18)).} The value $y$ will be called the {\it external colour\/} of $x$, denoted by $y=x^{\text{ext}}$. \enddefinition \definition {Precontours} For any configuration $x$ denote by $B(x)$ the collection of all its nonstratified (i.e\. stratified for {\it no\/} $y\in\es$) points. If $C$ is a {\it finite\/} connected component of $B(x)$ then the pair $$ \gb =(C,x_C) $$ will be called the {\it precontour of $x$\/} and we will write $ C=\supp \gb $. We will say that $\gb$ is a {\it precontour\/} if it is a precontour of {\it some\/} configuration $x \in \ex$. \enddefinition \definition {Admissible systems of precontours} By an {\it admissible system of precontours\/} we will mean any system $\Cal D =\{\Gamma _i\}$ of precontours which is a collection of {\it all\/} precontours of {\it some\/} configuration $x $. \footnote{In the next 50 pages, we will work exclusively with {\it finite\/} contour systems (in a finite volume $\Lambda$).}\enddefinition \remark{Note} The configuration $x$ is uniquely determined by $\Cal D $ only in all the horizontal planes intersecting $\Cal D $\,, otherwise it will be given by the boundary condition $y \in \es$ given outside of the volume in which we will be working. It will be \footnote{ This is a slight abuse of notations but we will see later that the geometrical notions defined below with the help of $x_{\Cal D }$ will not depend, in fact, on the ambiguity in the choice of $x_{\Cal D }$.} denoted by $x= x_{\Cal D,y}$ or (usually) shortly by $x = x_{\Cal D }$; the external colour $y$ of $x$ will be denoted also as $x^{\text{ext}}$ and the external colour of $x_{\Cal D}$ as $x^{\text{ext}}_{\Cal D}$. Thus, we represent each diluted configuration $x$ or $x_{\Lambda}$ as $x = (y, \Cal D)$ where $\Cal D$ is the system of precontours of $x$ or $x_{\Lambda}$ and $y$ is an additional information about the boundary condition on $\Lambda^c$ resp. on ``the value in infinity'' if we are in infinite volume. The value of $y$ will be usually clear from the context and therefore mostly omitted in the notations; we will often speak about a configuration $x = \Cal D$, etc. \endremark \vskip1mm \head 3. Contours \endhead One could characterize the \ps \ theory as a play with the concept of a ``contour''; the latter being understand as some barrier between different ``correct'' (stratified in the context of the presented paper) regimes. The very idea of the ``removal'' of a contour $\gb$ -- together with the comparison of the partition function of the original regime ``inside'' $\gb$ with the partition function of the new regime (which appears after the removal of $\gb$) plays a decisive role there; and this must be supplemented by suitable combinatorial bound for the number (something like $C^N$) of different shapes of contours of a given cardinality $N$. However, it would be sometimes problematic to realize the idea of a ``removal'' of $\gb$ for a precontour $\gb$ which has a topologically complicated shape. As an example of such complicated shapes one can have in mind something like a ``bumerang'' or, say, a ``ring'' over such a bumerang or ``fingers'' of one precontour penetrating the ``interiors'' of other precontours. Surely the concept of an interior of these objects deserves a more careful definition. However, precontours are {\it not suitable objects to do that\/}. One solution of this problem is the concept of a wall introduced in [D]. The idea of the Dobrushin wall suggests some {\it glueing of precontours together\/} as a means to clarify the concept of an interior of such a wall (which is defined as a suitable conglomerate of precontours). However, when glueing these topologically difficult objects together one has to keep in mind another highly desired property of contours which should be fulfilled -- namely their ``connectedness'' (whatever one could have in mind by this). The concept of a Dobrushin wall was used also by several other authors later, notably in \cite{HKZ} and \cite{HZ} (while in \cite{MS} the geometrical notions used for the description of the interface were more different). Such an approach resolves the problems mentioned above; however it seems to be applicable (by considering the projection of the situation appearing at the interface to the sublattice $\zet^{\nu -1}$) only in the special situation of {\it one\/} interface. This construction can be hardly transferred to the situations where two or more parallel ceilings appear, and therefore we will {\it not\/} follow such an approach. Of course, our abandoning of the concept of a wall is meant on technical level only, for topologically ``obscure'', nontypical representatives of the idea of a wall. Nevertheless the concept of a wall as a connected conglomerate of precontours defined with the help of a $\nu -1$ dimensional projection will be lost, replaced by a more general notion of a contour below. This will cause some problems, especially when the necessity check some ``connectedness'' of the contours will finally emerge in section III. Namely, we will have to control the convergence of cluster expansion formulas developed there; some combinatorial bounds for the number of possible contours (more generally, for the number of``clusters of contours'') will be necessary. Remember that precontours {\it are\/} connected by their very definition, but unfortunately there is {\it no\/} natural hierarchy between them. We will see later in section III that the problem of connectedness of our more general (``less connected'' than usually) contours can be managed with some effort. See the section ``Tight sets'', part III. There is no need to discuss this problem just {\it now\/} -- and so we give the definition of a contour below without explaining here the true advantages of such a choice. This will become clear later. In fact, we could live without contours in the forthcoming text, keeping only the notion of a compatible system of precontours and having in mind that operations like the ``removal'' of $\gb$ should be never applied to precontours or systems of precontours which are not compatible. However, the concept of an isolation introduced below {\it will\/} be extremely important for us and the notion of a contour is just its obvious byproduct, so why not to keep such a notion, traditional in the \ps \ theory \footnote{ In the previous versions of the paper we had a much more restrictive notion of a contour $\gb$ -- being defined as some conglomerate of precontours whose any subconglomerate is ``split'' (in the topological sense, in a suitable horizontally invariant section of $\gb$) by the rest of $\gb$. The reader will see in part III that our contours really are some ``connected conglomerates of precontours'', though in a different (and more general) sense than in \cite{D}. See the section ``Tight sets''.} also here. \definition {Contours} Say that a subset $S \subset T$ of $T \subset \zv$ is {\it isolated in\/} $T$ (or {\it from\/} $T\setminus S$) if $$ \dist ( S,T \setminus S) ) \geq \diam S.\tag 2.5 $$ By the {\it diameter of $S$\/} we will mean, everywhere in the following, the $l_{\infty}$ diameter of $S$. Say that $T$ is {\it tight\/} if it has no isolated subset. An admissible system $\gb =\{\gb_i\}$ of precontours will be called a {\it contour\/} if the set $ \cup_i \supp \gb_i$, denoted also as $\supp \gb $, is tight. Denote further by $\ext \Gammab$ the collection of all points of $(\supp \Gammab)^c $ which can be accessed from infinity by some vertically constant (i.e. keeping the last coordinate intact), correct (in the sense that each point of the path is a stratified one) connected path. Define also the volume (having only a minor importance in our approach!) $ V(\Gammab)=(\ext\Gammab)^c \,. $ We will say that $\gb$ is a contour {\it in a volume\/} $\Lambda$ if $\gb$ {\it is isolated from\/} $\Lambda^c$ i.e. if $\supp \gb$ is isolated from $\Lambda^c$. We will denote this fact by $\supp \gb \sqsubset \Lambda$ (we introduce such a special notation $ S \sqsubset \Lambda$ only for sets $S$ which are supports of contours) or $ \gb \sqsubset \Lambda$. So we have $$ \gb \sqsubset \Lambda \ \ \ \Longleftrightarrow \ \ \ \dist(\Lambda^c,\supp \gb)\geq \diam \supp \gb \ \ \ \ \Longleftrightarrow \ \ \ \vv \subset \Lambda \tag 2.6 $$ where the important notion of ``protecting zone'' $\vv$ of a contour $\gb$ is defined as \footnote{ However, if $\Cal D$ is an admissible system which is {\it not\/} a single contour then we put (compare (2.16)) $\vvd = \cup_i \vvi $ where $\{\gb_i\}$ is the representation (see Proposition below) of $\Cal D$ in terms of its contours.} $$ \vv = \{ t \in \zv:\ \dist(t,\supp \gb) \leq \diam \supp \gb\} .\tag 2.7$$ (We have $V(\gb) \subset \vv$, as simple inspection of any horizontal slice of $V(\gb)$ shows.) \enddefinition \definition{Admissible families of contours} A collection of contours $\Cal D =\{\gb_i\}$ will be called admissible if each $\supp \gb_i$ is isolated from \ \ $ \bigcup_{j \ne i: \diam \gb_i \leq \diam \gb_j } \supp \gb_j.$ Contours of the system $\Cal D$ which are isolated from the remainder will be called {\it interior\/} contours of $\Cal D$. \enddefinition \proclaim{Proposition} Any admissible system $\Cal D =\{ \gb_i\}$ of precontours can be uniquely decomposed to an admissible system $\tilde \Cal D =\{\tilde \gb_j\}$ of contours. \endproclaim This is immediate if we realize what the interior {\it contours\/} of $\Cal D$ are: they are defined as admissible subcollections of $\Cal D$ containing no smaller isolated subcollection. Notice that an intersection of any two isolated subcollections of $\Cal D$ is isolated resp. empty, if moreover one of these subcollections is minimal. Thus two different interior contours are always disjoint and mutually isolated. Having such (uniquely defined) interior contours we can remove them (replacing each such interior contour $\tilde \gb$ by its corresponding external colour) and one obtains a smaller admissible system where the new, ``bigger'' interior contours can be again uniquely determined etc. It is straightforward \footnote{ Notice that if an interior contour $\tilde \gb$ of $\Cal D$ with a smallest possible diameter is removed from $\Cal D$, then no interior contour $\tilde \gb'$ with a smaller diameter than $ \tilde \gb$ can be found in the remainder $\Cal D \setminus \gb$!} to establish the isolation properties of such a system. \remark{Notes} 0. We stress that ``admissibility'' for a system $\{\gb_i\}$ of precontours means just a {\it nonconflicting\/} prescription of ``colours outside of $\cup_i \supp \gb_i$ '' while admissibility of a system of {\it contours\/} requires also the isolation of any $\tilde \gb_i$ from all contours $\tilde \gb_j:\tilde \gb_i \prec \tilde \gb_j$. 1. The volume $V(\gb)$ is the intersection of all standard volumes containing $\supp\gb$. In the rest of the paper (the exception will be the proof of our Main Theorem; actually this will be really relevant only when developing some explanatory notes interpreting the detailed structure of the phases constructed by Main Theorem) we will work only with {\it finite\/}, often standard, volumes $\Lambda$. In fact, one could restrict oneself only to the {\it cubes\/} $\Lambda$ in most following considerations. The boundary condition $y \in \es$ will be always given on the boundary of $\Lambda^c$ . (Thus we will work with diluted configurations having a finite number of precontours only.) 2. Let us comment once again the topological problems we meet here, in comparison to the ordinary translation invariant \ps \ theory. In the ordinary translation invariant theory, the elements of the ``reference set'' $\es$ are constant configurations. In such a case one immediately realizes that precontours (called there contours) are ``crusted'', in the sense that the (only!) infinite component of $(\supp\Gamma )^{c}$ (which will be called the exterior of $\gb$) satisfies the property that all its points are ``$y$ -- correct'' where $y$ is the external colour of $\Gamma $. Even more importantly, the interior of $\gb$ (the union of the remaining components) is always ``disconnected'' from the exterior of $\gb$, for any precontour $\gb$. This enables to construct ``telescopic equations'' relating the diluted partition function in a given volume $\Lambda$ to the ``crystallic'' (see e.g. [S]) partition functions of the external contours appearing in $\Lambda$, and therefore to the diluted partition functions of the {\it interiors\/} of these contours. This is {\it not \/} so here, where precontours can have the shape of Dobrushin's ``walls'' separating various types of ``ceilings'' i.e. various horizontally invariant configurations in our case. Even if the notion of an exterior resp. interior of a contour is well defined these two volumes are usually not separated by an ``inpenetrable crust'' like in the ordinary translation invariant situations. \head 4. Expression of the Hamiltonian and the Peierls Condition \endhead For any stratified configuration $y\in\es$ \ we define now its ``density of energy'' at $t\in \zv $ : $$ e_t(y)=\sum_{A\ni t}\Phi _A(y_A) |A|^{-1} \,. \tag 2.8 $$ Given a (pre)contour $\gb$ one would like to define also a quantity having the meaning of the ``(pre)contour energy''. One could think, for {\it contours\/}, for example about the ``energy excess'' of $H(x_{\gb})$, where $x_{\gb}$ denotes \footnote{We might say that the information about a contour $\gb$ consists of 1) information about its support $\supp \gb$ and 2) information about a configuration $x_{\gb}$. Here, we need to know only the value of $x_{\gb}$ restricted to ${\supp \gb}$. However, quantities like $F(\gb)$ will require the knowledge of $x_{\gb}$ in the whole $\vv$.} the configuration having $\gb$ as its {\it only\/} contour, with respect to ``something like $H(x_{\ext})$'' where $x_{\ext}$ denotes the ``external colour of $\gb$''. However, such a straightforward approach to the definition of an energy of a contour is reasonable only in the very special cases (like the Ising model with zero magnetic field) when the density of energy inside $\gb$ is the {\it same\/} as outside of $\gb$, at {\it any} horizontal level. Otherwise it will be useful (to keep the interpretation of a contour energy as a quantity which is ``localized on $\supp \gb$ '') to replace the quantity $H(x_{\gb})-H(x_{\ext})$ just mentioned \footnote{This will be, of course, also {\it very\/} important later -- see (2.19)!} by the following quantity (perhaps too formally defined at first sight): Let us first extend the notation $e_t(x)$ for any (even {\it nonstratified\/} in $t$) $x$ by putting $$ e_t(x)=e_t(\hat x)\,; \quad t=(t_1,t_2,\dots,t_{\nu}) \tag 2.9 $$ where $\hat x$ is the {\it stratified continuation\/} of the vertical section $\{x_{(t_1,\dots,t_{\nu-1},(\cdot))} \}$. Now define, for any finite $G \subset \zv$ (or, more generally, such that any vertical section of $G$ has finite components only) and any diluted configuration $x$ such that all points of $G^c$ are {\it stratified\/} ones, the auxiliary configuration $x^{\text{best}}_G$ {\it minimizing\/} the sum in variable $z$ below, where $y$ denotes the ``external colour'' of $x$ (notice that the terms of the infinite sum below are equal to zero outside of $G$): $$ \sum_{t\in\zv}(e_t(z) -e_t(y)) \tag 2.9' $$ under the condition that the variable $z$ equals to $x$ on the set $G^c$ and also on the set \ $\partial_r G=\{t \in G,\ \dist(t,G^c)\leq r\}$. Write $x_{\gb}^{\text{best}}$ instead of $(x_{\gb})^{\text{best}}_{\supp \gb}$ where $x_{\gb} =x_{\gb,y}$ ($y$ is a boundary condition) is a configuration having $\gb$ as its single contour. \footnote{ One should not try to interpret such an ``artificially chosen'' configuration $x_G^{\text{best}}$ or $x_{\gb}^{\text{best}}$ in the points where it is {\it not\/} stratified. In particular the choice of $x=x_{\gb}^{\text{best}}$ should not be confused with the effort (natural perhaps, but inconvenient technically) to minimize $H(x)$ from (2.2) under the restrictions above!} Put $$ E (\gb)=H(x_{\gb})-\sum_{t\in\zv} e_t(x_{\gb}^{\text{best}}) \ \ \ \text{where} \ \ \ H(x) = \sum_{A \subset \zv} \Phi_{A}(x_A). \tag 2.10 $$ Of course this is again only a formal expression. However, the terms in the sums on the right hand side of the equation for $E(\gb)$ can be obviously reorganized such that a sum with only a finite number of nonzero terms is obtained. Namely one can rewrite (2.10) in the following equivalent and more precise way, pouze footnote (even for {\it pre\/}contours; but then (2.10) must be written more carefully, not using (generally nonexistent) $ y = x_{\gb}^{\text{ext}}$) $$ E(\gb) =\sum_{A \subset \supp \gb} \bigl( \Phi_A(\gb_A) - \sum_p \frac{|A\cap p|}{|A|} \ \Phi_A(( \hat x_{p}^{\text{best}})_A) \bigr) \tag 2.11 $$ where $\gb_A$ denotes the restriction of $\gb$ to $A$, the second sum is over all ``verticals'' $p =\{t_1,\dots,t_{\nu -1}, (\cdot)\}$ intersecting $\supp \gb$ and $\hat x_{p}^{\text{best}} $ denotes the horizontally invariant extension of the restriction of $x_{\gb}^{\text{best}}$ to $p$. In the (very special!) case of a {\it stratified\/} ``reference''configuration $x^{\text{best}}_{\gb}$ the formula (2.11) can be written in the following, more simple and more transparent way: $$ E(\gb) =\sum_{A \subset \supp \gb} \bigl( \Phi_A(\gb_A) - \Phi_A(( \hat x_{\gb}^{\text{best}})_A) \bigr). \tag 2.11' $$ Then we have the following expression of the Hamiltonian (2.2): \proclaim{Theorem 1} Let $x$ be a configuration such that all its (pre)contours have supports in a given finite \footnote{$\Lambda$ will always denote a finite set in the sequel. The constant $C$ in the formula ( 2.12) does not affect Gibbs measure defined by (2.12) in a fixed volume $\Lambda$. We will put $C =0$ in the following considerations.}set $\Lambda$. Then there is a constant $C =C(\Lambda,y), y = x^{\text{ext}}$ such that $$ H(x_\Lambda |x_{\Lambda^c})=\sum_{t\in\Lambda } \hat e_t(x) + \sum_{\gb \in \Cal D} E(\gb) +C= \sum_{t\in\Lambda } e_t(x^{\text{best}}_{\Cal D}) + \sum_{\gb \in \Cal D} E(\gb) + C \tag 2.12 $$ where $\Cal D$ denotes the system of all precontours of $x$ and where we are using also the notation $ \hat e_t(x)$ for $ e_t(x^{\text{best}}_{\Cal D})$; (see (2.9) and (2.9')) . \endproclaim One has to prove that (2.2) with the notations (2.9), (2.11) imply (2.12). Instead of giving here the details of the (noncomplicated!) arithmetics of this argument we notice only that (2.12) follows immediately from the formal expression (2.10). Notice also from (2.11) that $E(\gb)$ is a local quantity, depending on $\gb$ only and also an additive one: $$E(\gb\cup\gb')=E(\gb)+E(\gb') . \tag 2.13 $$ \definition {Peierls condition} In the following we will assume that for any finite $G \subset \zv$ and for any stratified configuration $y\in\es$ the following inequality holds with a large \footnote{The requirement on the largeness of $\tau$ depends on the dimension $\nu$. For translation invariant problems and $\nu =2 $, one could possibly reach something like $\tau > 5$. Unfortunately, our estimates will not be apparently always the strongest possible and our actual requirement on $\tau$ will be much stronger, something like, say, $\tau > 50$.} $\tau>0$ : $$ \sum \Sb \gb:\ \supp \gb=G \ \& \ x_{\gb}^{\ext}=y\endSb \exp(-E(\gb)) \leq \exp(-\tau|G|) \tag 2.14 $$ where the summation is over all {\it contours\/} having the support $G$. (We recall that we include the inverse temperature into the Hamiltonian and therefore $\tau$ is of the order of the inverse temperature in most examples.)\enddefinition \remark{Note} In practice, one can establish (2.14) with the help of inequalities, for some $\tau^* \geq \tau$, $$ E(\gb) \geq \tau^* |G|. \tag 2.15 $$ The belief is that (like in the usual translation invariant situations) such a condition {\it should hold in most usual situations\/}; in fact the counterexamples of (translation invariant) models with nonvalid Peierls condition constructed by Pecherski \cite{PECH} and later by Miekisz \cite{MIE} are rather nontrivial. Concerning an effective constructive criterion how to check the validity of (2.14) or (2.15), one could suggest a suitable modification of the Holsztynski -- Slawny criterion (see \cite{HS}, \cite{EFS}). This question surely deserves further study; specifically for stratified ground states. We did not investigated this in detail yet. \vskip1mm \head 5. Strictly Diluted Partition Functions. \endhead \vskip1mm Recall that in addition to the volume $V(\gb)$ we have another (technically much more important later) notion of $\vv$ which will be needed for any contour $\gb$ (and also for some special, ``recolorable'' {\it systems\/} of contours --see part III, sections 4 - 5.) The reason for replacing of $V(\gb)$ by $\vv$ in definitions like (3.19) will be understood only later. \definition{Strictly diluted partition functions $Z_{\updownarrow}^y(\Lambda)$} Recall that for a contour $\gb$, $\vv$ is the union of points from $\zv$ whose distance to $\supp \gb$ is not greater than $\diam \gb$. If $\Cal D = \{\gb_i\}$ is an admissible system of contours then we define the volume $\vvd$ as \footnote{ One could take a union of ``protecting zones'' $V'(\gb)$ of {\it pre\/}contours of $\Cal D$ here and everywhere. Once we decided to introduce the concept of a contour, such a modification would bring no advantage. Also, one could argue that the protecting zone $V'(\gb)$ is chosen unnecessarily big. However this will make no harm, too. In later sections of part III (namely in the construction of a ``skeleton'' of $\gb$) we will be a little bit more cautious when tackling ambiguities analogous to those appearing in (3.19). In fact, a neighborhood $V'$ of $V(\gb)$ up to a distance $\approx \log \diam V(\gb)$ would be sufficient here, too, but we do not care.} $$\vvd = \cup_i \vvi. \tag 2.16$$ We denote, from now on, by $ Z^y_{\updownarrow}(\Lambda)$ the ``strictly diluted'' partition function \footnote{{\it This\/} partition function will be our main object and we will study it systematically in the forthcoming text. In fact, other interesting partition functions $Z(\Lambda)$ (even those which can {\it not\/} be expressed solely in the language of contours) can be easily related to suitable ``strictly diluted'' partition functions $Z^y_{\updownarrow}(\Lambda')$ of subvolumes $\Lambda' \subset \Lambda$. It is also possible to modify the forthcoming constructions to the case of more general ``$y$ -- like'' boundary conditions; in the final section of the paper we will discuss modifications needed for the study of such more general partition functions. However, especiallly for the case of stable boundary conditions, this is a minor point of the whole story, and the difference between different partition functions mentioned above is unessential. On the other hand, strictly diluted partition functions are, of course, {\it not\/} suited to {\it directly\/} describe, without some adaptations, also the {\it nonstable behaviour\/} in a volume $\Lambda$ because very large (of the size comparable to the size of $\Lambda$) contours are {\it forbidden\/} here just by definition. This problem will not bother us in the present paper concentrated on the stability properties of various $y \in \es$.} $$ Z^y_{\updownarrow}(\Lambda )=\sum_{x_{\Lambda}} \exp(-H(x_\Lambda |y_{\Lambda ^c})) \tag 2.17 $$ where the sum is over all $x_{\Lambda}$ such that all contours $\gb$ of the configuration (on $\zv$) $x_{\Lambda} \cup y_{\Lambda^c}$ satisfy $ \dist(\supp \gb,\Lambda ^c)\geq \diam \supp \gb $; this condition will be written also as \footnote{This is just a suitable notation, like $A \cap \cap B = \emptyset$ below. In general, $A \sqcap B \ne B \sqcap A$ !} $$ \supp \gb \ \sqcap \ \Lambda^c = \emptyset \ \ \text{or} \ \ \supp \gb \sqsubset \Lambda. \tag 2.18 $$ In other words, we require that each contour $\gb$ of $x_{\Lambda}$ would be isolated from $\Lambda^c$. Let us introduce also another notation $\Cal D \ssubset \Lambda$, equivalently $\Cal D \cap \cap \Lambda^c = \emptyset$ used for {\it systems\/} $\Cal D$ of contours $\{\gb_i\}$ and defined by $$ \supp \Cal D \ssubset \Lambda \ \ \text{iff} \ \ \ \gb_i \sqsubset \Lambda . \tag 2.19$$ \enddefinition \remark{Note} For a comparison, introduce also the analogy of usual, ``diluted'' partition function $$ Z^y(\Lambda )=\sum_{x_{\Lambda}} \exp(-H(x_\Lambda |y_{\Lambda ^c})) $$ where the summation is over all (``diluted'') configurations, whose contours satisfy the requirement, say \footnote{ The precise requirement on the {\it boundary\/} condition is less important here; the real difference from usual ``diluted'' partition functions being in the changed meaning of our very notion of a contour.} $ \dist(V(\gb),\Lambda ^c)\geq 2 $. Such a traditional formulation of ``diluteness'' would be less convenient with respect to our approach and it will {\it not\/} be used later. \endremark The rest of this paper (and the essence of the Pirogov -- Sinai theory in its presented version) consists of our effort to {\it expand\/} the considered diluted partition functions (2.17) as far as it is possible or reasonable; then we deduce some important corollaries from them, like the estimate of the probability to find the external configuration $y$ in a given point of $\zv$, in the Gibbs state determined by the boundary condition $y \in \es$. Namely, the complementary event will be shwn to have a probability of the order $\exp(-C\tau)$. The attempt to expand partition function (2.17) can be based also on the older idea of a contour model \cite{PS} (or of a metastable contour model \cite{Z}). Though that notion in fact almost disappeared from the presented version of Pirogov -- Sinai theory (instead of speaking about ``metastable contour models'' we will work, in fact, only with {\it expansions\/} of the partition functions of the ``metastable submodels of the given model'') it is useful to start with some intuitive arguments suggesting the introduction of the basic notion of a {\it contour functional\/}. This is just the introduction to the later technical constructions. We will see that the very notion of a contour functional (\cite{S}) survived in our approach (in contrary to the idea of a contour model) and it is still, of course, of a central importance! \head 6. Abstract \ps \ Models and Their Gibbs States. \endhead \vskip1mm \definition{Strictly diluted configurations} By this we mean a configuration $x = (x^{\text{best}},\Cal D) $ where $\Cal D$ is an admissible contour system. If we are in a volume $\Lambda$ we require also that $\vv \subset \Lambda$ would hold for all the contours $\gb$ of $\Cal D$. More precisely, by a configuration we mean a pair $(x^{\text{best}}, \Cal D)$ where the first term $x^{\text{best}} = x^{\text{best}}_{\supp \Cal D}$ (recall that $x$ is stratified in all points of $(\supp \Cal D)^c$) contains both the information about the local ``colour''(the particular value from $\es$) of the boundary of contours as well as about the boundary condition on $\Lambda^c$; and $\Cal D$ is a ``matching compatible'' system of contours. Contours $\gb$ and their energies $E(\gb)$ are either defined as above from some ``physical model'' (2.2) or are given \footnote{ Such a preparation of the model (i.e. its conversion to the form (2.12) by a suitable definition of the notion of a (pre)contour) is not at all unique and in some cases, there is even no ``most natural way'' how to do it, for a given concrete model. One can adapt the general scheme given above in various ways for various concrete situations -- by modifying the concept of a ``stratified point'', for example, in broader terms even by defining contours as objects quite different from those we constructed here. Remember, for example, that in the theory of translation invariant low temperature Ising models, contours are traditionally defined as some selfavoiding {\it paths} (or surfaces in dimension $3$) defined by plaquettes from the the dual lattice. We leave to the reader to specify all the details of the analogy of an expression (2.12) in such a case (then $x^{\text{best}}$ is just a ``best possible'' {\it constant configuration\/} if translation invariant situation is considered) and also in the stratified case. However, these details are rather irrelevant for the general strategy of the theory.} as some abstract entities satisfying (2.14). In the latter case, only the {\it last relation\/} of (2.12) has sense, of course. Analogously, we can treat the densities $e_t(y)$ as some abstract local functions of $y \in \es$, having {\it no\/} apriori relation to the energies $E(\gb)$. Such an abstract approach will turn out to be very useful later and in the following, we will work {\it exclusively\/} \footnote{Except of some explanatory remarks interpreting the meaning of our Main Theorem, section III, 9.} with configurations allowing the expression (2.12) and satisfying (2.14) with a large $\tau$. It is generally advisable to develop the theory in such an abstract setting (2.12), with the Peierls condition (2.14) being established. The reformulation of the original model to the language (2.12) can be considered as a suitable ``preparation'' of the given ``physical'' model (in some situations, verification of the Peierls condition may be a nontrivial task!); the Pirogov -- Sinai theory actually {\it starts with the setting \/} (2.12) \ \& \ (2.14). \enddefinition \definition{ Gibbs states of abstract \ps \ models} In finite volumes, by an $y$--th Gibbs state on the space $\ex(\Lambda,y), y \in \es$ (of all $y$ -- strictly diluted configurations in $\Lambda$) we will mean the probability measure, denoted by $P_{\Lambda}^y $, determined by the hamiltonian (2.12) (recall that we can take $C=0$ there) with the partition function (2.17). \enddefinition What are infinite volume limits of such measures? (See also \cite{S} for an additional information -- for a case of contour models). Shortly speaking, take the following class of configurations defined on the whole volume $\zv$. Consider all the pairs $(x^{\text{best}}, \Cal D)$ where $x^{\text{best}}$ is defined as before and $\Cal D$ satisfies (e.g.) the following condition. \definition{Locally finite contour systems $\Cal D$ on $\zv$} Say that an admissible contour system $\Cal D$ is locally finite, resp. $I$ -- finite if for each $t \in \zv$ the collection of contours $\gb \in \Cal D$ such that $\gb$ is ``sufficiently close to $t$ '', in the sense that \footnote{ Of course there is some arbitrariness in such a requirement. ``Sufficiently close to $t$'' is our substitute for a more suspect (in the stratified case!) concept of a ``contour $\gb$ containing $t$ in its interior''.} $\gb \sqcap \{t\} \ne \emptyset$, is {\it finite\/}, resp. has a diameter at most $I(t)$. Take $I(t) = |t|$ (for example). Denote by $\ex^{\text{abstract}}_{ \text{fin}}$ the collection of all locally finite abstract \ps \ configurations. Denote by $X^{\text{abstract}}_{\Cal I} = \cup_y X^{\text{abstract}}_{\Cal I,y}$ the collection of configurations which are $\tilde I$ -- finite for some shift $\tilde I(t) = I(t+s)$ resp. also have an external colour $y$. Each set $X^{\text{abstract}}_{\Cal I,y}$ is thus the countable union of compact sets $X^{\text{abstract}}_{I,y}$ of $I$ -- finite $y$ -- like configurations. \enddefinition \definition{ Gibbs probabilities on $\ex^{\text{abstract}}_{\text{fin}}$} Any configuration space $\ex^{\text{abstract}}_{\Lambda,y}$ of all strictly diluted, with respect to the boundary condition $y \in \es$, configurations of the abstract \ps \ model in a volume $\Lambda$ can be naturally embedded into the space of all locally finite $y$ -- like configurations $\ex^{\text{abstract}}_{\text{fin}, y}$. In such a way, any probability measure $P^y_{\Lambda}$ can be viewed as a probability on $\ex^{\text{abstract}}_{\text{fin}, y}$. Introduce the $\sigma$ -- algebra of subsets of $\ex^{\text{abstract}}_{\text{fin}, y}$, generated by all ``cylindrical'' events $\{(x^{\text{best}},\Cal D): \gb \in \Cal D\}$ where $\gb$ is a contour system. \footnote{ Surely, this is not an exhaustive list of all cylidrical events on $\ex^{\text{abstract}}_{\text{fin}, y}$, if we consider this configurations space as a subset of the original configuration space $\ex$. However, for the Gibbs state on $\ex^{\text{abstract}}_{\text{fin}, y}$ one will see that the $\sigma$ -- algebra generated by ``cylindrical'' sets above, completed by sets of measure zero contains also all the usual cylindrical sets, like the specifications of spins in finite sets $ \Lambda \subset \zv$.} \enddefinition Consider now the thermodynamic limits of these measures. \definition{Thermodynamic limits and limit Gibbs states} By a thermodynamic limit over volumes $\Lambda \to \zv$ we mean just the very general requirement that we have a {\it suitable sequence of volumes\/} $\Lambda_n$ such that the relation the $\dist(t,\Lambda_n^c) \to \infty$ holds for some (equivalently, for each) $t \in \zv$. Any thermodynamic limit (taken in the above sense) of probabilities $P^y_{\Lambda}, \ y \in \es$, understood in the sense of {\it convergence on each cylindrical set\/} $\{(x^{\text{best}},\Cal D):\gb \in \Cal D\}$ will be called a {\it Gibbs probability\/} on $\ex^{\text{abstract}}_{\text{fin}}$, respectively on $\ex^{\text{abstract}}_{\text{fin},y} \subset \ex^{\text{abstract}}_{\text{fin}}$ if it exists as a $\sigma$ -- additive probability on the space $\ex^{\text{abstract}}_{\text{fin}}$, resp. even on $\ex^{\text{abstract}}_{\text{fin},y}$. The value $y \in \es$ will be called {\it stable\/} if the limit has support in $\ex^{\text{abstract}}_{\text{fin},y} \subset \ex^{\text{abstract}}_{\text{fin}}$ (and not in some other $\ex^{\text{abstract}}_{\text{fin},\tilde y}, \tilde y \in \es$). See section 9, part III for detailed information. \enddefinition \remark{Notes} 1. Some care concerning the ``allowed shape of $\Lambda$'' will be necessary -- to avoid possible entropic repulsion/attraction effects which could affect the picture. The use of so called ``conoidal`` sets (see the last section of the paper, boundaries of conoidal sets do not contain flat pieces) makes the above effects negligible in questions concerning the ``stability of a stable $y$ also inside of the volume $\Lambda$''. This is important when interpreting, in finite volumes, the contents of our Main Theorem. 2. We will not discuss here the details of the relation between the notion of a Gibbs measure on the usual configuration space $\ex$ (with Hamiltonian (2.2)) and Gibbs probabilities on $\ex^{\text{abstract}}_{\text{fin}}$. Some explanatory remarks will be given at the end of section III but the problem of the mutual relation between the usual notion of a Gibbs measure on $\ex$ and Gibbs measures on suitable spaces $\ex^{\text{abstract}}_{\text{fin},y}$ deserves an additional study (perhaps in a suitably more general setting covering also other applications of \ps \ theory).\endremark \vskip1mm \head 7. The idea of a contour functional \endhead The basic task of the Pirogov -- Sinai theory is to determine those configurations $y$ among the ``reference'' ones (reference means stratified in our case) which are stable in the sense that they give rise to a Gibbs state, whose almost all configurations are some ``local'' perturbations of the considered reference (stratified) configuration. (More precisely, they are ``$y$-- diluted'' almost surely.) A useful and intuitively appealing tool to determine whether a given configuration is ``stable'' is the construction of a ``metastable model'' \cite{Z} around the given reference configuration. To define such a metastable model one introduces (\cite{S}, \cite{Z}) an auxiliary quantity called the ``contour functional'' $F(\gb)$ which can be interpreted as the ``work needed to install the given contour'' and which is used for the {\it test\/} whether $\gb$ is allowed as an {\it external\/} contour of the metastable model. To get an idea of the value of such a testing quantity let us start with its simplified version, for the case when ``no contours within contours are allowed'': Put $$ F_0(\gb)=H(x_\gb)-H(x_\gb^{\ext})=E(\gb)-A_0(\gb) \tag 2.20 $$ where $$ A_0(\gb) = \sum_{t\in V(\gb)} (e_t(x_{\gb}^{\ext})-e_t(x_{\gb}^{\text{best}})). $$ This quantity is of course just a first approximation to the more relevant quantity given at this moment only formally by $$ F_{\text{formal}}(\gb)= \log Z^y_{\text{ref}}(\zv)-\log Z^{\gb}(\zv) \tag 2.21 $$ where $y$ is the external colour of $\gb \ \ (y=x_{\gb}^{\ext})$ and $Z^{\gb}$ denotes the partition function ``over all configurations on $\zv$ containing the contour $\gb$''. The ``reference'' partition function $Z^y_{\text{ref}}(\zv) $ is over all configurations having the ``colour'' $y$ on $\supp \gb$ (i.e. being $y$ stratified there) and satisfying moreover the property that their contours do not ``touch'' $\supp \gb$. Below we will define, by the relations (3.21) and (3.22), a {\it rigorous counterpart\/} of this quantity, which will play a very important role in the sequel. For contours which are ``not very big'' the quantity $F_0$ is a good approximation to $F_{\text{formal}}$. It enlightens somehow the concept of a small (or recolorable) contour used below; the term $A_0(\gb)$ typically satisfies an estimate like $$ A_0(\gb)\leq C|V(\gb)| \tag 2.22 $$ with $C$ much smaller than $\tau$ (imagine the Ising model with a small external field; then $C$ has the order of its intensity) and therefore if, say, $ C|V(\gb)|\leq \frac{\tau}{2}|\supp \gb| $ (this will surely hold for contours which are ``not too big'') we have, from (2.15), the inequality $$ F_0(\gb)\geq E(\gb)-C|V(\gb)|\geq \frac{\hat \tau}{2} |\supp \gb|\,. \tag 2.23 $$ We see that $y=x_{\gb}^{\ext}$ is really a ``local ground state'' because installing of a ``not too big'' contour {\it increases\/} its energy. Unfortunately, it is not at all trivial to define quantities like $F_{\text{formal}}$ rigorously in our situation. While analogous task is solved rather straightforwardly in other applications of the Pirogov -- Sinai theory (where contours are ``crusted'' in the sense that there is no dependence between events inside and outside $\gb$), here the presence of ``penetrable ceilings'' (flat horizontal parts of boundaries of $V(\gb)$ which do {\it not\/} belong to $\supp \gb$) causes problems! These problems however can be solved with the help of suitable {\it expansions\/} of the model, and this is the main subject of the forthcoming part of the paper. \vskip1mm \head 8. A summary of the main result \endhead Let us summarize once more our basic constructions applied so far and outline our basic result. Consider a model (2.2). For any standard finite volume $\Lambda$ , rewrite the Hamiltonian $H$ according to the formula (2.12), using the ``single spin interactions'' $e_t(y),y \in \es$ (more generally $\hat e_t(x) =e_t(x^{\text{best}}), x \in \ex$) and the ``contour energies'' $E(\gb)$. \footnote{Recall that while $e_t$ are the ``densities of energy of $x$ at $t$'', the more general auxiliary quantities $\hat e_t(x)$ do not have such a nice interpretation. However, they enable us to formulate the Peierls condition in a useful and general way (2.14).} \definition{An Outline of Main Theorem} Assume that a Hamiltonian (2.2) is given satisfying the Peierls condition (2.14) when reformulated by (2.12). Then one can construct quantities $$h_t(y) = e_t(y) - s_t(y) $$ such that in any volume $\Lambda$, for any boundary condition $y \in \es$ given outside of $\Lambda$, the probability $P^y_{\Lambda} [\gb]$ that a contour system $\gb$ appears in $\Lambda$ is given by the formula \footnote{We will return to (2.24) by formulating its precise analogoue (3.93) at the end of the paper.} $$ P^y[\gb] = \exp (-\tilde F(\gb)) \ \ \ , \ \ \ \tilde F(\gb) = E(\gb) - \sum_{t \in \Lambda} (h_t(y) -h_t(z)) + \Delta \tag 2.24 $$ where $z$ denotes the local ``colour'' (value from $\es$) induced by $\gb$ outside of $\supp \gb$ and where the correction term $ \Delta = \Delta (\gb, \Lambda,y)) $ is usually {\it small\/}. More precisely the bounds $\Delta \leq \varepsilon |\supp \gb|$ respectively $|\Delta| \leq \varepsilon |\supp \gb|$ hold in the case when $\gb$ is strictly diluted in $\Lambda$ and $y$ is ``not too unstable'' (see section 8, part III for a more precise description) respectively (the stronger bound) if moreover no ``residual'' matching compatible contours could be placed outside of $\gb$. The quantity $h_t(y)$ may be interpreted as the ``density of the free energy, at $t$, of the $y$ -- th metastable state'' and it satisfies the following property: Whenever a choice of $y$ is made such that $h_t(y)$ has its {\it ground value\/} (in the sense that the sum $\sum_{t \in \zet}h_t(y)$ cannot be lowered if $y$ is changed on a strip of a finite width) then the {\it existence of a Gibbs state\/} $P^y$ {\it of the ``$y$ --th type'' is guaranteed\/}. This will be called the {\it stability\/} of $y$. By the ``$y$ --th type'' we mean that $P^y$ almost any configuration looks, roughly speaking, like an ``infinite sea of $y$ with small islands of perturbations''. The terms $s_t(y)$ disappear in the zero temperature limit $E(\gb) = \infty$. \footnote{ Recall once again that we include the inverse temperature into $H$ and therefore all our quantities $E(\gb)$ (and also $e_t(y)$!) are actually proportional to the inverse temperature $\beta = T^{-1}$. However, the {\it difference\/}, for $t$ fixed, between various relevant $e_t(y),y \in \es $ is typically of much smaller order than $e_t(y)$.} They are given by suitable cluster expansion formulas, which can be written as follows, according to the number $k$ of contours forming a cluster $\{\gb_i ,i =1,\dots,k\}$: $$s_t(y) = \sum_k s^k_t(y) \ \ \ \text{where}\ \ \ s_t^1(y) =-\sum_{\gb: t \in \supp \gb} |\supp \gb|^{-1} \exp (-F(\gb)) \tag 2.25 $$ the summation being over all ``recolorable'' contours $\gb$. The property of recolorability will be defined in part III; all contours will be automatically recolorable up to a certain size; this size will depend on the external ``colour'' $y$ of $\gb $ and will become {\it infinite\/} for $y$ stable. The contour functional $F(\gb)$ mentioned in the previous section has a value {\it very close\/} to the (nonlocal) value $\tilde F(\gb)$ in (2.24); see however (3.22) for an exact formula showing also the {\it locality\/} of this notion and see again (2.21) for its intuitive meaning. The remaining terms $s_t^2(y),\dots$ can be expressed analogously as sums $$ s_t^k(y) = \sum_{\{\gb_i\}: \cup_i \supp \gb_i \owns t} n_{\{\gb_i\}} \prod_{i=1}^k \exp(-F(\gb_i)) \tag 2.25'$$ over suitable ``clusters'' \ $\{\gb_i\ ;\ i =1,\dots,k\}$ \ of recolorable contours mentioned in (2.25). The combinatorial coefficients $n_{\{\gb_i\}}$ do not depend, once $\gb_i$ are recolorable, on the Hamiltonian $H$. They are described below in the proof of Main Theorem \footnote{The reader will realize later that the terms forming the sum $s_t^k(y)$ are rather complicated but really {\it very\/} quickly decaying -- the leading term in these sums being $\exp(-C\tau N)$ where $N$ is the sum of cardinalities of the supports of contours of a smallest possible cluster appearing in the sum (2.26).}. One could modify the expressions (2.25), (2.25') by allowing sums over {\it all\/} $\gb$ (including the {\it non\/}recolorable ones) with suitably defined, {\it large and fixed\/}, value of $F(\gb)$; see \cite{Z} for the concept of the ``truncated functional'' $\hat F(\gb)$. Then all the terms $s_t^k(y)$ would change continuously (with some additional care they would change even differentiably or even more smoothly; see \cite{ZA}) and moreover it can be proven that they satisfy (if $F(\gb)/ |\supp \gb|$ is large!) the Lipschitz condition, with a small constant, with respect to the parameters $e_t(y)$ and $E(\gb)$ of the Hamiltonian (2.12). Finally the partition function $Z^y_{\updownarrow \ \text{meta}}(\Lambda)$ corresponding to all diluted ``metastable'' configurations in $\Lambda$ (see part III, section 9; the subscript ``meta'' means {\it all\/} configurations if $y$ is stable, otherwise some configurations -- namely those with ``external nonrecolorable contours''-- are excluded) can be expressed via cluster expansion terms mentioned above, for any standard volume $\Lambda$ as follows : $$ \log Z^y_{\updownarrow \ \text{meta}}(\Lambda) = -\sum_{t \in \Lambda} e_t(y) + \sum_{\{\gb_1,\dots,\gb_k\}} n_{\{\gb_i\}} \prod_{i =1,\dots,k} \exp(-F(\gb_i)) = \tag 2.26 $$ $$ = -\sum_{t \in \Lambda} h_t(y) + \Delta(\Lambda) \ \ \ \text{where} \ \ |\Delta(\Lambda)| \leq \varepsilon |\partial \Lambda| \tag 2.26' $$ where the sum is over all clusters of recolorable contours $\gb_i \sqsubset \Lambda$ and $\varepsilon$ is small. \footnote{Again (like in (2.24)), in comparison to the translation invariant situations the last estimate in (2.26') is not very strong and will {\it not\/} be used below. Namely in general we may have $|\partial \Lambda| \gg |\supp \gb| $ for a Dobrushin wall $\gb$ such that $V(\gb) = \Lambda$, and then (2.26') is useless. The expression by cluster expansion in the first part of (2.26) will be much stronger and more useful in such situations.} \enddefinition \remark{Notes} 1. Apparently, there are {\it no other\/} stratified Gibbs states than those corresponding to the ``minimal values of $\sum_{t_{\nu}} h_{t_{\nu}}$''; however this is not proven in this paper. 2. The functions $h_t(y)$ can be made in fact infinitely differentiable if defined more carefully; even local analyticity can be achieved, like in \cite{ZA}, on the manifolds of parameters where the collection of configurations $y$ ``minimizing the value $\sum_{t_{\nu}} h_{t_{\nu}}(y)$'' is fixed. We plan a separate paper devoted to both these questions. \endremark \vskip1mm \head 9. Application: Stabilization of one dimensional model in three dimensions \endhead It will be useful to formulate the following corollary of our Main theorem which is better suited to concrete applications (see the paper in preparation \cite{EMZ}). Reasonable formulation of such a result can be given only in the class of {\it infinite range\/} Hamiltonians and we refer to part III for more details concerning the (rather straightforward!) generalization of our Main Theorem to the models with exponentially decaying interactions. \definition{Essentially one dimensional interactions} A horizontally invariant, infinite range, indexed by squares from $\zv$ interaction $\{\Phi_{\square}(x_{\square})\}$ which is nonzero only if $x_{\square}$ is a restriction of some {\it stratified\/} configuration will be called essentially one dimensional. It can be identified with an interaction acting on $S^{\zet}$ and indexed by intervals from $\zet$. One should distinguish {\it two variants\/} of such a notion formulated 1) in the language of the abstract \ps \ model resp. 2) in the language of the model (2.2). Below we use the (suitably adapted) {\it first\/} approach which is much more straightforward once we have based our exposition on the expression (2.12). Namely will moreover assume that the interactions $\Phi_{\square}$ act only {\it outside of all contours\/}, and that they even may have a suitably modified value (compared to the usual one, which is horizontally translation invariant) in a distance, say, less than $\diam \square$ from some contour of the configuration. Of course, for applications to real {\it spin\/} model (2.2) one needs the approach 2) above -- allowing $\Phi_{\square}$ to act, in a horizontally translation invariant way, also {\it inside\/} of the contours -- if the restriction to $\square$ of the given configuration of contours is stratified. This is commented in more detail in section 8, after the proof of Main Theorem. A more detailed exposition of the implication \ \ Main Theorem $\Rightarrow$ Corollary \ is needed (it is easier in the simplified interpretation of $\Phi_{\square}$ we are using here) see the end of Section III and the paper \cite{EMZ}. \enddefinition \proclaim{Corollary} Consider a model (2.12) with a hamiltonian, denoted by $H_0$, satisfying the Peierls condition (2.14). Consider further a whole class of models on the same configuration space $S^{\zv}$ with a Hamiltonian $$ H = H_0 + H_1 \tag 2.27 $$ where $ H_1 = H^{\varepsilon,\omega}_1 $ is from the family $\Cal H^{\varepsilon,\omega} $ of all essentially one dimensional Hamiltonians defined by interactions $\{\Phi_{\square} \}$ acting outside of the contours of $H_0$ and decaying like $$ |\Phi_{\square} (x_{\square})| \leq \ \varepsilon \ \omega^{\ \diam \square} , \tag 2.28$$ where $\omega, \varepsilon$ are chosen such that $ \omega < 1$ and ${\varepsilon}/{1 -\omega} \ll 1 $. Consider the (uniquely defined!) decomposition of the Hamiltonian $H_0$ into its essentially one dimensional \footnote{ If we have in mind stratified $y$ only.} ``ground part'' and the ``Peierls part'': $$ H_0 = H_G + H_P $$ where \ \ $e^{H_P}_t(y) \equiv 0$ for all $y$ and $E^{H_G}(\gb) \equiv 0$ for all $\gb$. Then there is a mapping \footnote{ We are not interested here at all in the Peierls part of the image. Namely, when finding ground states among various $y \in \es$, the knowledge of the Peierls part (satisfying (2.14)) of the Hamiltonian is irrelevant.} $$ \{ H \equiv H_G +H_P + H_1 \mapsto H_G + H_1 +\tilde H_1 \}:H_0 + \Cal H^{\varepsilon,\omega} \mapsto H_G + \Cal H^{\tilde \varepsilon,\tilde \omega} \tag 2.29 $$ where $\tilde \omega = \max(\omega,C\exp(-\tau))$ with suitable $C = C(\nu)$ \footnote{That is, $\tilde \omega = \omega$ if $\omega$ was not chosen ``excessively small with respect to $\exp(-\tau)$.} and $\varepsilon < \tilde \varepsilon < \varepsilon + C \exp(-\tau)$ (the same constant $C$) having the following properties:\roster \item It is Lipschitz continuous in the sense that whenever both $H_1,H_1'$ are from (2.27) such that $H_1 - H_1'$ is from a class $\Cal H^{\varepsilon', \omega}$, $\varepsilon' < \varepsilon$, then $\tilde H_1 - \tilde H_1'$ is from $\Cal H^{\varepsilon'', \omega}$ where $\varepsilon'' = \varepsilon' \exp(-C'\tau) $ with another constant $C' = C'(\nu) $. \item If $y$ is a ground state of $H_G +H_1 +\tilde H_1$, i.e. if it minimizes the sum $\sum_t \tilde h_t(y)$, where $\tilde h$ denotes the density of free energy of the model with Hamiltonian $H_G +H_1 + \tilde H_1$, then the stratified $y$--like Gibbs state of the Hamiltonian $H= H_G +H_1 +H_P$ exists. \footnote{Ground state of $H_G +H_1 +\tilde H_1$, not of $ H$ resp. $H_G + H_1$! And we are looking for Gibbs states of $H$.} \item In particular, if we choose $\varepsilon' $ such that $\varepsilon - \varepsilon' > C \exp(-\tau)$ then there is \footnote{ By the inversion mapping theorem.} for any $\tilde H \in \Cal H^{\varepsilon',\omega}$ a preimage $ H = H_G +H_P +H_1, \ H_1 \in \Cal H^{\varepsilon, \omega}$ such that $H_1 +\tilde H_1 = \tilde H$ and such that there is one to one correspondence between the stratified ground states of $H_G +\tilde H$ and the stratified Gibbs states \footnote { More precisely we mean here only the stratified Gibbs states {\it constructed by our Main Theorem\/}. However, we do not expect an existence of any other stratified Gibbs states of $H$.} of the Hamiltonian $H$. \endroster \endproclaim \remark{Notes on the proof} See section 8 and also \cite{EMZ} for more information. The interpretation of the additional interaction $\tilde H_1$ in (2.29) is straightforward if we start with a abstract \ps \ model ($H_1 = 0$) in (2.27). Then the cubic interaction $\Phi_{\square}$ for the new hamiltonian $ \tilde H_1$ arises very naturally; namely it suffices to perform a partial summation, over cubes, in the formulas $$\log Z^y_{\updownarrow}(\Lambda)= -\sum_{t \in \Lambda} e_t(y) +\sum_T k_T(y) \ \tag 2.30$$ and then to use (3.1). We actually have $k_T =n_T \prod_i \exp(-F(\gb_i)$) with some combinatorial coefficients $n_T$, and the sum is over all clusters $T =\{\gb_i\}$ of ``recolorable'' contours with the external colour $y \in \es$. The partial summation is taken over all the cluster expansion terms $k_T(y)$ which can be ``packed'' into a given cube $\square$ but not into a smaller cube. If we start with a nontrivial $H^{\omega,\varepsilon}$ then the situation is analogous; see section 8 and \cite{EMZ} for some arguments; the aplication of Corollary above to {\it spin\/} models (not only to models already formulated in the canonical form (2.12)) requires some additional discussion. \endremark \head III. The Concept of a Mixed (Partially Expanded) Model. Recoloring. \endhead \vskip1mm This is the final and the main part of the paper. It is devoted to the construction of suitable {\it expansions\/} of partition functions of models considered in part II. Here we introduce the important technical notion of a ``mixed'' (or, partially expanded) model which serves as an {\it intermediate construction\/} between the original concept of an ``abstract Pirogov -- Sinai model'' and our final aim which is an utmost expansion of the partition functions of the considered model. Cluster expansions were always an important tool in the Pirogov -- Sinai theory. However, in previous versions of this theory, the expansions were viewed merely as some auxiliary technique applied to the study of special polymer models (contour models) which were constructed first. One could think that the cluster expansion method could be replaced by ``something else'' giving ``comparably nice'' expressions (or, possibly, suitable bounds only) for the partition functions of the contour models. This is not so here where the idea of a partial expansion enters even our basic terminology, namely the concept of the mixed model. For example an analogy of the important notion of a ``metastable model'' (see [Z]) can {\it not\/} be apparently even defined here without the language of expansions; also the very formulation of our Main Theorem is based on this language. Of course the idea of a ``partial expansion'' is not at all new. It was used (in various context also in situations close to the subject of the presented paper - see \cite{I}, \cite{HKZ}, \cite{B},\dots by many authors mostly as an important but auxiliary tool, while in our formulation it is really the cornerstone of our theory. Our basic expansion step (Theorem 3, Theorems 4 and 5) -- called recoloring by us {\it incorporates\/} some of the usual cluster expansion ideology (based on the expansion by power series) into the very construction of the contour functional. As a consequence, our use of cluster expansion technique is {\it selfcontained\/} and we need no references to the literature. We can, however, mention \cite{M}, \cite{KP}, \cite{DZ} (as the papers having direct influence on the present paper) from the numerous literature on the subject of cluster expansions. The construction of one ``recoloring step'' (Theorem 4 and Theorem 5) \footnote {In fact, one could formulate directly a version of Theorem 4 recoloring all the shifts of $\gb$ at once i.e. giving directly Theorem 5. In future expositions of our method we plan to follow this more direct approach.} will not yet give the required expansion of the model. This step must be {\it repeated\/} many times (infinitely many in the thermodynamic limit). The iterative nature of our constructions cannot be hidden ``somewhere into the proofs'' but appears already at the level of the basic notions. We organize this part of the paper as follows. In section 1 we analyze the notion of a ``cluster'' (of supports of contours or, more generally, of another clusters); the clusters are then identified as suitable trees mapped to $\zv$. Then, in section 2, we define the central concept of a {\it mixed\/} model. This notion corresponds to an idea of a ``partially expanded model''; however it is useful to consider such a concept in a broader sense. Section 3 describes an important procedure -- the``recoloring'' (i.e. the removal of $\gb$ from the model $\&$ adjustment of the new cluster series such that the partition functions would not change) of an interior contour $\gb$, in the context of a general mixed model. The important concept of a recolorable contour is introduced here: it corresponds, roughly speaking, to the validity of the Peierls condition for the contour functional $ F(\gb)$. Section 4 applies the result of section 3 in such a way that recoloring of all the shifts of $\gb$ is obtained; the resulting new mixed model is again a horizontally translation invariant one if the original mixed model satisfied this property. Technically, sections 3 and 4 (complemented by later sections 6 and 7) form the core of our paper. Later sections 5, 6, 7 are then devoted to the problems of the succesive construction of ``more expanded'' mixed models: An important intermediate result is Theorem 6 (section 6) giving a sufficient condition for the recolorability of an interior contour system in a general mixed model. Namely, to have more specific examples of recolorable contour systems we introduce there a related but better controllable notion of a {\it small\/}, more precisely {\it extremally small\/} contour system which is more useful than a (more general) notion of a recolorable contour system. The message of the sections 5 to 7 is roughly speaking the following: once there are some small contour systems in the mixed model then there is still ``something left to recolor'', i.e there are still some recolorable interior contour systems in the model. The notion of an extremally small contour system is an elaboration of the older idea of a ``small'' or ``stable'' contour (\cite {Z}). Notice that small resp. extremally small contour systems can contain some ``large'' (``not extremally small'') contour systems as their ``interior'' contour subsystems. Theorem 6 is proved with the help of Theorem 7; the latter is actually some general statement about the ``connectivity constant'' of some special (``tight'') sets appearing in the study of extremally small contour systems. Only after finishing the sequence of all the expansions (recolorings) organized by Theorem 5 we will be able to say what the {\it metastable\/} model is -- in Section 8. This will be defined as a submodel of the original abstract \ps\ model where only those configurations containing {\it no\/} ``residual \footnote{ ``Residual'' means surviving in the ``totally expanded'' model: by the totally expanded model we mean the final mixed model remaining at the moment when the inductive procedure of its partial expansions was completed. See Section 8 below.} external contours'' will be admitted! Section 8 formulates then our main result, using the quantities called ``metastable free energies'' just constructed by expansions. It turns out that that the minimality of the metastable free energy of some $y \in \es$ really means that there are {\it no contours at all\/} in the totally expanded model under such a boundary condition i.e. the metastable model corresponding to $y$ gives an appropriate $y $-- th {\it Gibbs state\/}. The fact that under ``stable'' boundary conditions, ``everything is recolorable'' (i. e. a complete expansion of the partition functions is obtained) is the core of the proof of Main Theorem. Having proved the preparatory Theorems 5, 6 this will be almost a tautology. Our new method based on Theorem 5 and Theorem 6 replaces the previous coarser arguments from \cite{Z} which cannot be used in these new situations. However, even in the situation of \cite{Z} our new method is simpler (at least conceptionally) and more powerful: the {\it bounds\/} for the partition functions employed in \cite{Z} are now systematically replaced by statements on {\it quick convergence of the corresponding expansions\/}. See \cite{ZRO}. We plan to show the advantages of this new approach in the study of further situations which are not covered by the usual variants of the Pirogov -- Sinai theory. \head 1. Clusters \endhead \vskip1mm This section prepares some technical notions and constructions needed for the proper formulation of the expansions which are used below. Cluster expansions of partition functions of polymer models are usually written, in the literature, in the following form: $$ \log Z^y(\Lambda ) =- \sum_{t \in \Lambda} e_t(y) +\sum _{T \subset \Lambda} k_{T}(y). \tag 3.0$$ Actually, we will be interested here in the expansions of the partition functions $\log Z^y_{\updownarrow} (\Lambda)$ where $y \in \es$ and the condition $T \subset \Lambda$ will have a more restrictive meaning, see below in (3.20). \footnote{ We will have always $\dist (S,\Lambda^c) \geq \diam S$ for any set $S $ used in the construction of the cluster $T$.} The quantities \ $ k_T(y), \ T \subset \zv$, are some local functions \footnote{ Actually, they will be given as products of exponentials of contour functionals, like in (2.26). See the forthcoming sections for the extensive discussion of the values $k_T$.} dependent on $T$ and on the ``local colour''$y$ (in fact on $y_{\vvt}$, see below). They are indexed by ``connected''(in some generalized sense) clusters $T$ (see below for more details about this notion) and they are ``quickly decaying'' e.g. like (this will be the form used by us) $$ | k_{T}(y)| \leq \varepsilon^{\conn T} \tag 3.1 $$ where $ \conn T $ is something like the ``cardinality of a minimal commensurately connected set containing the cluster $ T $''. See Definition 4 below for the definition of the quantity $\conn T $ which will be used in our later considerations. The constant $\varepsilon > 0$ will be asssumed to be {\it sufficiently small\/}, often it will be of the order $\varepsilon = \exp (- C\tau)$ for a suitable constant $C$, This will be our {\it final goal\/}: establishing of such expansions for a collection -- as large as possible -- of diluted partition functions of the given model. More complete information will say that the quantities $ k_{T} $ are in fact sums of quantities indexed by some ``clusters of sets (resp. of contours) '' (and having a value which is a $\pm$ product, over the cluster, of contour functionals $\exp(-F(\gb_i))$) having the given support $ T $. While one can ignore the detailed description of the structure of $k_T$ when applying the above expressions (e.g. in order to obtain useful {\it bounds\/} for partition functions -- and this was the typical application of the cluster expansions in most previous variants of the \ps \ theory) here it will be necessary to retain the more precise information because these expansions will be iterated repeatedly many times. Before defining the notion of a cluster formally, we start with the explanation of the notion of $ \conn T $ for the case when $ T $ is a {\it set\/}. Our definition relates such a notion to finding of some shortest ``commensurately connected'' superset containing $T$; this will be important later in this section when analogous construction will be applied also to a general cluster. We start in fact with the definition of a closely related quantity denoted by $ \Conn T $ which is defined in a more direct way. In the definition of $\conn T$ (see below) we will use the notion of an abstract {\it tree\/}, sometimes also with a specified {\it root\/}: \definition{Abstract trees} An abstract tree is defined as an equivalence class, with respect to isomorphisms of graphs, of nonoriented (binary) graphs without cycles. (By a cycle of a graph $G$ we mean a collection $\{\{t_1,t_2\},\dots,\{t_n,t_1\}\}, t_n \ne t_2$ composed of bonds of $G$.) If we wish to specify also the {\it root\/} of such a tree (i.e. mark one vertex of the graph), then such a tree can be concisely desribed also in a {\it recursive\/} way, just by specifying the collection of all subtrees, with marked roots, emerging if the root of the tree is removed. \enddefinition \remark {Note} Below, an identification of a cluster with a suitable tree will be given. This suggests that the following idea of the summation over cluster expansion series will be developed below: Summation over clusters will be reformulated as summation over trees, and instead of estimating the number of various clusters with the same length we will rather employ the idea of the {\it recursive\/} summation over all trees, based on the successive summation over the {\it outer\/} bonds of considered trees. It seems that this method gives good estimates and it also offers possibilities of a generalization to other interesting models. Therefore, we are developing this method here, in spite of the fact that the treatment given below is maybe somewhere too much detailed for the purposes of the forthcoming text. Namely, a weaker version could be made, which would be close in its spirit to our later approach of section 7 (based on the notion of a tight set; see the proof of Theorem 7). Nevertheless, we follow the method of summation over trees here, considering it also as a suitable reference for possible further applications of the method. (A summation over trees is used in the paper [CONZ] which is under preparation.) \endremark \definition{Commensurate trees on $\zv$} By a {\it commensurate tree $\Cal T$ on $\zv$\/} we will mean the object satisfying the following three requirements: \roster \item It is a pair $\Cal T = (G,\phi)$ consisting of {\it an abstract tree \/} $G$ and a {\it mapping\/} $\phi$ of this abstract tree $G$ to $\zv$. Such a mapping can be constructed, after fixing the root of the given abstract tree, in a recursive way (following the recursive definition of an abstract tree above): simply the image of the newly added root is specified at each stage of the recursive construction. The vertices of the abstract tree $G$ are mapped (generally not one to one) onto some subset of $\zv$ which will be called the {\it support\/} of the given commensurate tree and denoted by $\supp \Cal T$. Notice that several vertices of the given abstract tree $G$ can be mapped to the same $t\in\zv$. \item The {\it bonds\/} of the tree $\Cal T$ (more precisely, the corresponding images of bonds of $G$ in $\zv$) constructed in (1) are always (unordered) pairs of the following special type: $$ \{t, s\}\, ; s = t + 2^k\vec e_i $$ where $ k\in\en\, , t \in 2^k \zv\, $ and where $\, \vec e_i\, $ is either zero or a vector of the canonical base of $\zv$. More precisely we consider all the bonds $\{t=\phi(A), s=\phi(B)\}$ which are images under $\phi$ of the corresponding bonds $\{A,B\}$ of the abstract tree $G$. (We put no limitations on the number of such bonds per a given pair $\{t,s\}$.) \item The {\it commensurability\/} of $\Cal T$ is meant in the following sense: if $\{\phi(A), \phi(B)\}$ and \ $\{ \phi(A), \phi(C)\}$ are two neighboring bonds of the tree $\Cal T$ then $$ {1 \over 2} \rho(\phi(A),\phi(B)) \leq \rho(\phi(A), \phi(C)) \leq 2 \rho(\phi(A),\phi( B))\, \, \tag 3.2 $$ where the distance $\rho(\phi(A),\phi(B))$ between $\phi(A) =t $ and $\phi(B) =s $ is defined as $2^k$ resp $1$ according to whether $s=t+2^k \vec e_i$ or $s=t$ in the above relation. \item We define the length of such a tree as the total number of its bonds {\it excluding\/} all the bonds (``loops'') of the type $\{\phi(A) = t, \phi(B) =t\}$. \endroster \enddefinition \remark{Note} The usage of the lattices $2^k\zv$ and our very notion of a commensurability will be quite important in the following. The choice of the factor $2$ in (3.2) is more or less arbitrary but convenient later. We should notice that later, in the proof of Theorem 6 below, the notion of commensurability will be transcripted to an {\it alternate\/} language based on the usage of the unit {\it cubes\/} from lattices $2^k \zv$ (considered as cubes from the original lattice $\zv$) instead of the employment of the {\it bonds\/} of the type above. \endremark \definition{The quantities $ \operatorname{Conn T}$ and $ \operatorname{conn T}$} Given any set $T\subset\zv$ we assign to it a shortest possible commensurate tree containing for any $t\in T$ {\it at least one bond\/} of the type $\{t, t\}$. We will denote one such tree (it is often not determined uniquely, even if its root is already selected) as $\Cal T= \Cal T(T) \, $. We recall that the length of the tree was defined by (4) above and therefore the ``loops'' (bond of the type $\{t,t\}$) are not contributing to the length of the tree; the condition that all such loops are in the considered tree can be replaced by requiring that any $t\in T$ belongs to some bond $\{A,B\}$ of $\Cal T$ having the length $\rho(A,B) \leq 2$. Define the auxiliary quantity $\Conn T$ as the {\it length\/} (see the point (4) above) of the above tree $\Cal T(T)$. In the following, it will be more useful to have a modified version of this quantity, denoted by $\conn T$ and defined as follows: $$ \conn T = \Conn T +[3 \nu\log_2\diam T] + 6 \nu \tag 3.4 $$ \enddefinition \remark{Note} (3.4) will be a more adequate quantity than $\Conn T$ in what follows; see Proposition below. Namely, the clusters of sets will be defined below in a recursive way as collections of objects (sets or contour systems) having the property that their diameter is not smaller than their distance to some other (bigger) object of the collection''; and when constructing additional commensurate path connecting a given set $T$ with a point in distance $\diam T$ one requires an additional amount of \ $\approx \log_2 \diam T$ steps: \endremark \proclaim {Lemma 1} Let $\rho(t, s) = d$. Then there is a commensurate {\it path\/} starting by the bond of the type $\{t, t\}$ and ending by $\{s, s\}$ having the length at most $[\,3 \nu \ log_2 \ d\, ]$. \endproclaim \demo{Proof of Lemma 1} It follows easily from the following considerations: first notice that it suffices to consider the case of the dimension $\nu = 1$. Consider now the path on $\zet$ with steps having the lengths (we start and end with loops but we do not count them, $k \geq 1$) $$ 1, 2, 4, \dots, 2^k, \dots, 4, 2, 1$$ which overcomes the distance $d= 3\cdot 2^k -4$. The length of this path is $2 k - 1 $. If $$ 3 \cdot 2^k - 4< d' <5 \cdot 2^k -4$$ then it is possible to construct a commensurate path overcoming the distance $d'$ simply by {\it doubling\/} some of the steps in the first half (including the middle step) of this sequence. Thus we need at most $2 k - 1 + k+1 \, \leq\, 3 \,[\log_2 d'] \, $ steps to overcome any distance $d'$ from the interval $[2^k,2^{k+1})$ (even from the interval $[0, \ 5 \cdot 2^k -4)$) which completes the proof for $\nu =1 $. The multidimensional case is analogous, just construct $\nu$ paths in each coordinate axis and intertwine their steps together suitably. \enddemo Now we come to the definition of a {\it cluster\/}. The notion of a cluster of subsets \footnote{ Actually, only {\it some \/} subsets of $\zv$ will be employed in the construction of a cluster. Namely the supports of so called ``recolorable'' contours or contour systems, see below in sections 3 -- 5.} of $\zv$ is defined recursively, retaining the letter T for the notation of clusters, as follows: \definition{Clusters of sets $\supp \gb$} \roster \item "{i)}" Any set $$ T = \supp \gb$$ where $\gb$ is a contour or a contour system (``recolorable'' one, see below) is a cluster. \item "ii)" If $T_i$ are some clusters, $T_i \ne T_j$ for $i \ne j$ and $T_0$ is from i) such that \footnote{We will consider in what follows only the ``standard'' clusters (see below). Then we will have actually a stronger condition $\supp T_i \cap \cap (\supp T_0)^c \ne \emptyset$; see section 3 of part III.} $$ \dist ( T_0, \supp T_i) \leq 4 \ \text{min}\ \{ \diam(\supp T_0), \diam (\supp T_i)\} \tag 3.5 $$ holds for each $i\geq 1$, then the pair $ T= (T _0, \{T_i\})$ is again a cluster. We denote by $ \supp T = T_0 \cup \cup_i \supp T_i\,\,$. The set $T_0$ will be called the {\it core\/} \footnote{The fact that we have {\it one\/} core and not, say, multiple cores consisting of several horizontal shifts of the same $T_0$ (which is related to the fact that we employ the ordering $\prec$ and not $\prec \prec$ in the definition of a cluster) is related to our method, based on the ``lexicographical'' Theorems 3 and 4. Clusters with multiple cores (and no need for the lexicographical order) would appear if these theorems would be reformulated for {\it several\/} copies of $\gb$ {\it at once\/}. We plan to prefer such an approach in the future.} of $T$. The external colour, let us denote it by $y$, of $\gb$ will be called the {\it colour\/} of the cluster $T$ and we will assume that $T_i$ already have the {\it same\/} colour $y \in \es$ \footnote{Of course there is, as always, an ambiquity in the extension of $y$ to the whole $\zv$.}. \endroster \enddefinition \remark{Note} The condition (3.5) is a technical one; its adequateness (with respect to our actual constructions) will be seen later in Theorem 4. The additional ``logdiam'' term added in the definition of $\conn T$ (in comparison to $\Conn T$) will be seen to be related to our very formulation of the condition (3.5). See Proposition below. \definition{A tree associated to a cluster} We assign, to any cluster $T$, a commensurate tree $\Cal T$ defined as follows:\ If the trees \ $\Cal T_0$ and $\Cal T_i$ were already constructed by definition above and by the induction assumption for $T_i$ \ (recall that $\Conn T_0 = |\Cal T_0| $ where $\Cal T_0$ is a shortest commensurately connected tree whose support contains $T_0$) then we define $\Cal T$ as the shortest possible commensurately connected tree containing, as mutually disjoint subtrees (with disjoint bonds $\{A,B\}$) all the trees \ $\Cal T_0$ and \ $\Cal T_i$ and such that all the components of \ $\Cal T \setminus \Cal T_0$ start with some loop of the type $\{t,t\}, t \in T_0$. To have an idea about the length of $\Cal T$ consider the following particular construction of a tree whose length should be ``close'' to that of $\Cal T$. Moreover one has a clearer idea about its shape, see below in (3.6)). $$ \Cal T' =\Cal T_0\, \cup \, \cup_i (\Cal T_i \cup P_i)$$ where $P_i$ are some shortest possible commensurate paths, starting in some loop $\{s_i, s'_i\}$ of $\Cal T_0$ and ending in some loop $\{t_i, t'_i\}$ of $\Cal T_i$. In analogy to the case of a set $T$, the quantity $\Conn T$ is now defined as the length of the tree $\Cal T $ and we put (compare (3.4)) $$ \conn T = \Conn T + [ 3 \nu \log_2 \diam \supp T] + 6\nu . \tag 3.4' $$ \enddefinition In order to reconstruct back the original cluster $T$ from a given tree $\Cal T$ the following notion of a {\it standard cluster\/} will be useful: \definition{Partial ordering of finite subsets of $\zv$} Fix the partial ordering $\prec$ on the collection $\Cal F$ of all finite subsets of $\zv$ resp. (this will be used later, starting from Theorem 5) the partial ordering $\Cal A \prec \prec \Cal B$ between the equivalence classes $\Cal A =\{A' = A + t, t \in \zv\}$ (the ``factorization of $\prec$ with respect to all shifts in $\zv$'') given by the following requirements: The ordering $\prec \prec$ extends the partial ordering $\diam A < \diam B$ (recall that we use the $l_{\infty}$ norm on $\zv$ everywhere) and it is invariant with respect to all shifts in $\zv$. The ordering $\prec$ moreover extends the ordering $A \prec A'$ iff $ A'= A -t$, $t \prec_l 0$ where $\prec_l$ denotes the {\it lexicographical ordering\/} on $\zv$. \enddefinition \definition{Total ordering} One can, and will extend the partial ordering $\prec $ (analogously, the partial ordering $\prec \prec$) to a {\it total\/} ordering on $\Cal F$ (resp. on its factorization w.r. to all shifts). \enddefinition \remark{Note} There is, of course, still some arbitrariness in such an extension. For example, one can add, to the requirements 1) and 2), also the following two requirements: 3) $A \subsetneq \tilde B \Rightarrow A \prec B$, where $\tilde B$ denotes again a suitable shift of $B$ and even, say, \footnote{ This will be convenient later, when defining {\it external contours\/} (in section 9), though this notion will be only of a marginal importance to us.} another relation \ \ 4) $ \vva \subsetneq V_{\updownarrow}(\tilde B) \Rightarrow A \prec \prec B $ if the former is valid for {\it some\/} shift $\tilde B = B +t$ and if we denote by $ \vva = \{ t \in \zv:\ \dist(t,A) \leq \diam A\} $. (This temporary notation, used only here should not be confused with the notion of $\vvd = \cup_i \vvi$ defined before for $\Cal D =\{\gb_i\}$.) This is natural and {\it consistent\/}, as an inspection shows, with the requirements on $\prec$ resp. $\prec \prec$ given above. We can say that $A \prec \prec B$ holds if at least {\it one\/} of the relations $ A \subset \tilde B, \diam A \leq \diam B, \vva \subset V_{\updownarrow}(\tilde B)$ ($\tilde B$ is suitable shift of $B$) is not equal to $=$. \endremark Before coming to the following definition, let us fix the {\it total\/} ordering $\prec$ defined above. Note that while the choice of the extension of $\prec$ played no role in the definition of contours and their admissibility, the fact that we have a total ordering {\it will\/} be important now. \definition{Standard clusters} Say that a cluster $ T= (T _0, \{T_i\})$ is a {\it standard\/} one if $S \prec T_0$ holds for any set $S$ used (as a core) in the recurrent definition of $T_i$. \enddefinition Notice that since $\Conn T $ is not greater than $|\Cal T'|$ we have the inequality $$ \conn T \leq \Conn T_0 + \sum_i \Conn T_i + \sum_i l_i + [ 3 \nu\log_2 \diam \supp T ] + 6\nu \tag 3.6$$ where $l_i$ are the lengths of the paths $P_i$ used in the definition of $\Cal T'$. \footnote{ In fact, in Theorem 6 we will show that for all clusters considered later by us, the quantities $\conn T$ and $|\supp T|$ will be of the {\it same order\/}. Moreover, one could apparently rewrite the present section for this (narrower) setting in the spirit analogous to that of later Theorem 6, without employing the bothering (but small !) logdiam terms. We prefer the more general exposition here in view of a wider applicability of the estimates obtained here also to other situations.} The following estimate will be used later in Theorems 4 and 5 (though in a slightly changed form). It says that having established a slightly stronger version of the estimate (3.1) for {\it sets\/} $ T$ one obtains (3.1) also for all {\it clusters\/} $T$ if the quantities $k_T$ are given by the recurrent formulas below. The quantity $\Conn T$ does not seem to have comparably nice properties; the additional `` logdiam'' term in our definition of $\conn T$ seems to be essential for our ability to give a recurrent proof of (3.9). Next we formulate two auxiliary results: The first one will be directly used later (in a slightly different form not changing its essence -- see the proof of Theorem 5). On the other hand, the second result is its corollary which we formulate in a {\it more general\/} setting (Theorem 2) which will be possibly interesting also in other situations where our method can be applied. This latter result resembles the classical Mayer method (see \cite{R1}). \proclaim{Proposition 1} Assume that the quantities $k_T$ are defined recursively by formulas $$ k_T = k_{T_0} \prod_i k_{T_i} \ \ \text{whenever} \ \ T =(T_0,\{T_i\}). \tag 3.7 $$ Assume that for the {\it sets\/} $T_0 \subset \zv$ the following stronger variant of (3.1) is valid: $$ | k_{T_0} | \leq \varepsilon ^{\Conn T_0 + 6 \nu(\log_2 (\diam T_0 + 4)+2)} . \tag 3.8 $$ Then the estimate $$ |k_T| \leq \varepsilon ^{\conn T} \tag 3.9 $$ holds also for all the clusters $ T = \{T_0,\{T_i\}\}$, with $\conn T$ defined by the preceding definition, assuming that it is already valid for all clusters $T_i \ ;\ i \ne 0$, in (3.7). \endproclaim \demo{Proof} We have to prove that $$ \Conn T_0 + 6 \nu \log_2 (\diam T_0 +4)+12\nu + \sum_i \conn T_i \geq \conn T . \tag 3.10 $$ Notice first the following simple estimate. Define the support $\supp T$ and the diameter $\diam T$ of a cluster $T=(T_0,\{T_i\})$ recursively by putting $\supp T = \supp T_0 \cup \supp T_i$ and $\diam T = \diam \supp T$. \proclaim{Lemma 2} Let $T_j$ be the longest of all clusters $T_i$ (maximizing the diameter). Then $$ \log_2 \diam T \leq \log_2 (\diam T_0 +4) + \log_2\diam T_j.$$ \endproclaim \demo{Proof of Lemma 2} This is straightforward: notice that the condition (3.5) implies the bound $$ \diam T \leq \diam T_0 + (1+1+1+1) \diam T_j . $$ Then we use the inequality $\log_2 (x + 4y) \leq \log_2 (x+4) + \log_2 y$ which is surely valid for $ x \geq 1 \text{ and } y \geq 1$. \enddemo The idea of the proof of (3.10) now is to reduce it to (3.10'') below, and to ``feed'' the necessary increment of $\conn T -\sum_i \conn T_i -\Conn T_0$ by the ``superfluous logdiam term'' in (3.8) resp. by the value $\conn T_j -\Conn T_j$ of the longest cluster $T_j$ (if it is much bigger than $T_0$). Notice that it suffices to establish the following bound, from which the required bound (3.10) is obtained by summing it with $3\times$ the bound of Lemma 2. $$ \Conn T_0 + 3 \nu\log_2 (\diam T_0 +4) + 6 \nu + \sum_{i \neq j} \conn T_i + \Conn T_j \geq \Conn T \tag 3.10' $$ Really, this is surely valid because we can rewrite it (notice that $l_j \leq 3 \nu\log_2 (\diam T_0 ) + 6\nu$ by Lemma 1 !) in a stronger form $$ \Conn T_0 + \sum_i (\Conn T_i + l_i) \geq \Conn T \tag 3.10'' $$ where $l_i \leq \conn T_i - \Conn T_i$ denotes the length of the path $P_i$ used in the definition of the auxiliary tree $\Cal T'$ (see Definition 4). Namely, the bound $ l_j \leq 3 \nu \log_2 \diam T_0 +6\nu$ \ follows from the condition (3.5) and Lemma 1. The validity of the last inequality (3.10'') follows from the very definition of $\Conn T$ (see (3.4), (3.6)) and this completes the proof of Proposition 1. \enddemo \definition{Notations} In the following we will usually write, for clusters $T$, $ t\in T, T\subset \Lambda, \dots$ instead of the more precise notations $t\in \supp T, \supp T \subset \Lambda, \dots $. By writing $G \in T$ we will mark the situation when the set $G=\supp \gb $ was used in the recursive definition of $T$ (as the ``core'' of some intermediate cluster used during the construction) of the cluster $T$. We will also extend the notation $\sqsubset$, more generally $\subset\subset$ to {\it clusters\/} (compare (2.19), but here we have a cluster instead of a single contour system) $$ T \ssubset \Lambda \ \ \text{iff} \ \ G \subset \subset \Lambda \ \text{for each}\ G \in T. \tag 3.11$$ \enddefinition Finally we have (see below for the proof) one important consequence of the condition (3.1), to be used later in the formulation of our main result. \proclaim {Proposition 2} If there is a small $\varepsilon$ such that for each cluster $T$ and each $y \in \es$, $$ |k_T(y)| \leq \varepsilon^{\conn T} $$ then the cluster series with the terms $k_T$ quickly converge in the following sense: for any $t \in \zet$, any $y \in \es$ and for any $d \in \en$ we have $$ \sum_{T: \ t \in T \& \ \Conn T \geq d} |k_T(y)| \leq (C\varepsilon)^d \tag 3.12 $$ and analogously for the condition $\conn T \geq d$ replaced (e.g.) by $|\supp T| \geq d$. \endproclaim We require such a bound only for the (``more natural'') quantity $\Conn T$. Recall that our introduction of the quantity $\conn T$ was motivated by the necessity to derive (3.1) for all {\it clusters} from something like (3.8) which should be assumed to be valid for all {\it sets} $T$. Once we {\it have\/} (3.1) for {\it all clusters\/} we can forget the quantity $\conn T$ and replace it by $\Conn T$ if the convergence of the cluster expansion is investigated. We will prove Proposition 2 in a broader setting, related to usual estimates in the theory of the Mayer expansions (see the book [R1]). It is easy to understand that Theorem 2 below actually {\it generalizes\/} Proposition 2 above: Below we identify any cluster $T$ with some commensurate tree $\Cal T$ on $\zv$. The number of standard clusters corresponding to $\Cal T$ will be shown to be at most $6^{|\Cal T|}$. Having this in mind the forthcoming Theorem 2 can be formulated for quantities $ k_{\Cal T}$ indexed by commensurate {\it trees\/} $\Cal T$ on $\zv$ instead of clusters. \definition{Reconstruction of the cluster $T$ from $\Cal T (T)$} \enddefinition The mapping from clusters to trees constructed above is not one to one. First notice that given a ``connecting commensurate tree'' $\Cal T$ on a {\it set\/} of cardinality $n$, there are no more than $2^n$ possibilities how to recognize the original set $T$ (if there is some) just by specifying the points from the support of $\Cal T$ not belonging to $T$. Second, assume that clusters are already constructed (recurrently) from {\it trees\/} instead of sets. How many ways are there then to reconstruct the trees $\Cal T_0$ etc. from $\Cal T$ ? Assign to any bond of $\Cal T$ a ``colour'' red, blue or grey such that grey subtrees are interpreted as the ``building blocks'' $T_0$ and $ T_i$ while red bonds denote the connecting subtrees -- except of the bonds ``ending the connecting subtrees, going from $\Cal T_0$ to $\Cal T_i$'' ($\Cal T_i$ were constructed in the previous step of the recurrent definition of $\Cal T$ ); these will be marked blue. (Analogous coloring is used inside of each $\Cal T_i$, by induction assumption.) Look at the biggest (in $\prec$) grey component; this must be $\Cal T_0$ and analogously, taking red paths with blue ends starting from $\Cal T_0$ one recognizes the roots of the subtrees corresponding to $\Cal T_i$ etc. Thus, one has a bound (surely a very crude one !) $6^n$ for the number of standard clusters $T$ with the same tree $\Cal T$ of the cardinality $n$. In the forthcoming applications, {\it all\/} the clusters constructed by us will be standard ones and so we will not discuss the possible modifications of the estimates given above which would be needed if also nonstandard clusters would appear. The following general result can be apparently useful also in other situations (see \cite{CONZ}); it is probably a ``folklore'' but we do not know a suitable reference. \proclaim{Theorem 2} Let the quantities $k_{\Cal T}$ be given as products of some quantities denoted by $k_b$ or $k_{\{t,s\}}$ (see the commentary below) \footnote{Recall that the ``bonds'' $\{A,B\}$ of an abstract tree $G$ are mapped by $\phi$ to unoriented pairs of points $\{t =\phi(A),s =\phi(B)\}$ from $\zv$. The notation $k_b$ is used instead of a more explicit notation $k_{\{t,s\}}$ for $b =\{t=\phi(A),s=\phi(B)\}$.} $$ k_{\Cal T} = \prod_b k_{\,b} $$ where the product is over all the ``bonds'' $b = \{\{\phi(A),\phi(B)\}\}$ of the commensurate tree $\Cal T =\{G,\phi\}$. Assume that the quantities $k_b$ are nonnegative and $k_{\{t,t\}} = 1 $ for each $t$. Let for any unordered pair $ b = \{t,s\} $ we have the estimate $$\sum k_{\,b'} \leq q \tag 3.13 $$ where the summation in (3.13) is over all unordered pairs $ b' =\{s,u\}, s \ne u $ which are commensurate with $b$ and $q$ is some small, e.g. $q < 1/4$, positive constant. Then for any pair $b$ we have the following bound for the sum over all commensurate trees $\Cal T$ containing the ``bond'' $b$ as its ``extremal bond'' and having the length at least $2$: $$ \sum k_{\Cal T} \leq k_b \cdot q' , \tag 3.14$$ where $q'$ can be chosen as $q' =3q$ (more precise estimate of $q'$ is given below). By the extremality of \ $b = \{t =\phi(A),s =\phi(B)\}$ we mean that one vertex of the pair $ \{A,B\}$ remains ``free'' (i.e. it is ``endvertex'' in the original abstract tree. \endproclaim \remark{Note} One can optimize the value of $q'$. A better value $q'_{\text{new}}$ of $q'$ can be sometimes found from the equation (see the end of the proof below) $$ \exp( q'') = 1 +q'_{\text{new}} \tag 3.14'$$ where $q''$ denotes the supremum, over all bonds $b$, of the sums on the left hand side of (3.13) but with modified terms $$ k_{b'}''= { k_{b'}(1+q')\over 1- k_{b'}(1+q')} $$ and where $q'$ is the previously established value in (3.14). One can further elaborate the process of finding the optimal value of $q'$. Namely, if all $k_b$ are small and if we have already established the smallness of $q'$ then we have the crude bound, say $k_{b'}'' < 2 k_{b'} (1+q')$ and hence we have also the inequality $q'' < 2q(1+q')$, We will see below in the proof of theorem that the solution $q'$ of the equation $\exp(q'') = 1 + q'$ can be actually placed into (3.14); it clearly minorizes the value $q'_{\text{new}}$ above. See below for more information on this argument. \endremark \demo{Proof} Apply the method of induction. Denote by $\sum^{ 1$ different types of bonds $b'$ stick to $b$; each of them has a multiplicity $l \geq 1$ and each such bond $b'$ is the starting bond of some (empty or nonempty) subtree, or $l$ subtrees. (Contributions of these situations were just counted.) We get the final estimate (notice that now we have the term $m!$ here) $$ \sum^{ 0\,$ such that for any $T$, $$ |k_T|\leq \varepsilon ^{\,\conn T} . \tag 3.17 $$ \endroster \remark{Notes} {\bf 1.} We will not study too much relations between models in {\it different\/} volumes $\Lambda$. In particular, we will not look for analogies of ``telescopic equations'' usually formulated for diluted partition functions (see \cite{S}). We will {\it not\/} need this. \newline {\bf 2.} For all the configurations $(x,\{\gb_i\})$ considered below (everywhere in what precedes our Main Theorem, section 8) we will have $x = x^{\text{best}}_{\{\gb_i\}}$ (see the discussion of this notion at (2.9)). In fact we consider the concept of a mixed model as a generalization \footnote{ It is important to notice that until section 8 we will assume {\it no fuctional dependence\/} of the quantities $k_T(y)$ on the values $E(\gb)$ (and $e_t(y)$). Such dependence will be studied only later, in the formulation of Main Theorem, when very concrete mixed models will appear as partial expansions of the original model.} of the concept of an abstract \ps \ model, contours $\gb$ being some ``abstract connected objects satisfying the Peierls condition'' while spin configurations will appear only as ``ground configurations'' $x^{\text{best}}$. Below we will usually omit the superscript ``best''. \newline {\bf 3. } We will later glue together, in our ``recoloring procedure'', some contour systems $\gb$ and some clusters $T$ such that $\dist(T, \supp\gb)\leq \diam T$; to form new clusters of some new mixed model. This is one of the reasons why we added, in the preceding section, the ``safety constant'' $3 \nu \log_2 \diam T$ to the quantity $\Conn T$ in the definition of $\conn T$ in order to keep the control over the connectivity properties of the new clusters formed by such (recursive) procedure. \newline {\bf 4. } The collection of allowed contours (more generally of allowed {\it configurations\/}, see below) may vary from one mixed model to another. Typically the allowed set of contours will be some {\it subset\/} of the original collection of contours (of some given abstract Pirogov -- Sinai model), and this subset will become even {\it smaller\/} when applying further expansions (recolorings) to the given model. On the contrary, the collection of nonzero $k_T$ will always {\it grow\/} with such an expansion. See the forthcoming section for more details. \newline {\bf 5. } The restriction of the assumption (3.16) to nonzero products $k_T\, k_{UT}$ is related to the fact that, in the forthcoming section, we will work, temporarily, with horizontally translation noninvariant models. In fact the new cluster quantities $k_T$ will be constructed successively in the lexicographic order, through an infinite sequence of intermediate (noninvariant) mixed models. If the condition (3.16) is complemented by the assumption that {\it both\/} $k_T$ and $k_{UT}$ are nonzero if at least {\it one\/} of them is nonzero and if the configuration space is horizontally invariant (in the sense of what contour systems are allowed in $\zv$) we will speak about the {\it translation invariant\/} mixed model. \newline {\bf 6. } Having specified the {\it collection of allowed precontours\/} of the mixed model we do not require, in principle, that {\it all\/} admissible collections of allowed precontours are allowed configurations of the mixed model. We are quite general at this point and at the moment we impose no special requirements on what collections of precontours are really allowed in our model. See forthcoming sections 3, 4, 7 for a more concrete information on the actual choice of the configuration space. \endremark The {\it partition function\/} of the mixed model will be always the {\it strictly\/} diluted one : $$ Z^{\alpha} _{\updownarrow} (\Lambda)= \sum_{\Cal D \ssubset \Lambda} Z^{\alpha }_{\Cal D } (\Lambda ) \tag 3.18 $$ where $\alpha $ is a boundary condition on $\partial_r ( \Lambda^c)$ (from $\es$) and $\Cal D $ is a contour system. The sum is over all contour systems $\Cal D$ such that $\Cal D \ssubset \Lambda$. Recall that the notation $\Cal D \ssubset \Lambda $, analogously $T\ssubset \Lambda$ means that the condition $\gb \sqsubset \Lambda$ \ i.e.\ $\vv \cap \Lambda^c = \emptyset$ more generally $\gb \ssubset \Lambda$ for a contour {\it system\/} is satisfied for any contour $\gb$ of $\Cal D $ resp. any contour system $\gb \in T$. Using the hamiltonian (3.15) we define $$ Z^{\alpha }_{\Cal D }(\Lambda )= \exp(-\sum_i E(\gb_i)) \exp(-\sum_{t\in\Lambda }e_t(\delta )) \exp(\sum_{T\ssubset \Lambda \setminus\supp\Cal D }k_T(\delta)) \tag 3.19 $$ where $\delta$, such that $ \delta^{\text{ext}} = \alpha $ denotes the configuration (see (2.9)) $\delta =(x_{ \Cal D })_{\supp \Cal D}^{\text{best}}$. Recall that it is stratified outside of $ \supp \Cal D$. \remark{Note} It seems unnatural to use the symbol $Z$ \ for the ``mere Gibbs factor'' (3.19). However, the case $k_T = 0$ is {\it not\/} the most characteristic example here. In a more general case, considered mixed model corresponds actually to some partial {\it expansion\/} of the model (2.18). Then (3.19) is really some partition function, corresponding to an event ``$\Cal D$ is the collection of (still) nonexpanded contours of the original model (2.18)''. \endremark \vskip1mm \head 3. Recoloring of an internal contour system $\gb$ \endhead This is a central construction of our approach, {\it replacing\/} (together with the forthcoming constructions of later sections) the concept of a (metastable) {\it contour model\/} used in the previous versions of the Pirogov -- Sinai theory. In this section we describe some abstract, ``algebraic'' aspects of {\it one recoloring step\/} (of an arbitrary mixed model). An invariant (conserving the horizontal invariancy) application of this construction will be given in the forthcoming section 4. A suitable {\it sequence\/} of (translation invariant) recoloring steps, yielding as its final result the ``total'' expansion of a given Pirogov -- Sinai abstract model will be discussed later, starting from Section 5. Such a recoloring will be just one step towards the desired ``total expansion'' of the model, and this step is described in detail in Theorem 3 and Theorem 4 below. Roughly speaking, recoloring of a contour (resp. of a contour system) $\gb$ will just mean a {\it replacement\/} of a given mixed model by another mixed model where $\gb$ will {\it not\/} already be allowed as a contour (system) and where some {\it new\/} quantities $k_T$ (for some {\it new} clusters $T$ containing $G =\supp \gb$ as its core) will appear. The remainder of the model will be kept intact and the crucial fact will be that the diluted partition functions (2.18) of both the original and recolored model will be required to be the {\it same\/} for all finite volumes. Let us start with the definition of the following important quantity $A(\gb)$ which ``measures the instability of $x_{\gb}^{\ext}$ in $V(\gb)$'' and which will play a key role later when defining a rigorous substitute for (2.21); see also the remark below. There will be several variants of the quantity $A(\gb)$ -- see below -- and the technical difference between their definitions (essentially the decision what volume will be used instead of $V(\gb)$) will be quite important in this section, in spite of the fact that the values of all these quantities will be roughly the same. Recall (it will be quite indispensable in what follows) the definition of a ``protecting zone over $V(\gb)$'' namely the notion of $\vv$. Recall that $ \vvd = \cup_i \vvi $. Recall the notations for clusters $T$ and sets $\Lambda$: $$ T \ccap \Lambda^c = \emptyset \ \ \Leftrightarrow\ \ T \ssubset \Lambda \ \ \text{iff} \ \ \vvg \subset \Lambda \ \ \text{for each} \ \ G =\supp \gb \in T . \tag 3.20 $$ \definition{The quantity $A(\gb)$} For any contour system $\gb$ denote by $A(\gb)$ the quantity \footnote {More precisely $ A(\gb,x|{\vv})$. For typhographical reasons, we denote here the restriction of $x$ to $V$ by $x |V$ instead of the usual $ x_{V}$. This configuration will be always given from the context and this justifies the shortened notation $A(\gb)$. Notice also that the choice of the volume (where the clusters $T$ live) in the sums above is somehow arbitrary. See below (especially at (3.25)) for the discussion of the other possible alternatives to $\vv$. Finally, it is only a matter of arbitrariness to require $T \ssubset $ and not $T \subset$ here.} $$ A(\gb,x|\vv) = \sum_{t\in V(\gb)}(e_t(y)- e_t(x)) + \sum\Sb T\ssubset \vv\setminus \supp\gb \endSb k_T(x)- \sum_{T\ssubset \vv}k_T(y). \tag 3.21 $$ Recall that here $x=(x_{ \gb})^{\text{best}}$ is (see (2.9)) the configuration minimizing the Hamiltonian $H(x)$ under the conditions $x_{\partial_r \supp \gb} = \partial_r \gb \ (= \{\gb_t, t \in \partial_r \supp \gb \})$, $x_{(V(\gb))^c} = y$, $y=x_{\gb}^{\ext}$. \enddefinition \remark{Note} Thus, to be able to determine $A(\gb)$ exactly we must know the configuration $x$ on the whole volume $\vv$ because $k_T(x)$ resp\. $k_T(y)$ depend on the values of $x$ resp\. $y$ on the set $\cup_{G \in T} \vvg$. However, the value of $x$ on $\vv$ will be normally determined by the context in which the contour system $\gb$ will appear and so we will often use the shorter notation $A(\gb)$ instead of the more precise notation $A(\gb,x|\vv)$ without any ambiguity. \endremark We will see that the quantity (see (2.11), (2.12) for the definition of $E(\gb)$) $$ F(\gb)=E(\gb)-A(\gb) \tag 3.22 $$ is a useful exact substitute for the formal quantity $F_{\text{formal}}$ from (2.21). The choice of the set $\vv$ will guarantee (among other convenient properties) that the clusters constructed below having the ``core'' $\gb$ will be sufficiently ``tight''. (This would {\it not\/} be the case if we would take mere set $V(\gb)$ there). However, if $\gb$ is an ``interior contour system'' of some bigger admissible collection $\gb\&\Cal D $ of all precontours of some configuration $(x_{\Lambda }, \gb\&\Cal D )$ in a finite volume $\Lambda$, the following modifications of the quantity $A(\gb)$ could be considered too and we mention them for comparison and better understanding of the nature of $A(\gb)$. We will be interested below only in the case when $\vv \cap \Cal D = \emptyset $ i.e. when $\gb$ is not ``tightly attached'' to $\Cal D$. \definition{$A_{\text{full}}(\gb)$ and other variants of $A(\gb)$} In analogy to (3.21) define also the modified quantities $A_{\loc}(\gb)$,\ $A_{\full}(\gb)$, $A_{\full,\Cal D, \Lambda }(\gb)$\ as in the relation (3.21) but with the volume $\vv$ in the second and the third sum on the right hand side of (3.21) being replaced successively by volumes $V(\gb)$,\ $\zv$,\ $\Lambda \setminus \supp\Cal D $. \ Corespondingly define, by (3.22), the quantities $F_{\loc}(\gb)$,\ $F_{\full }(\gb)$,\ $F_{\full,\Cal D , \Lambda }(\gb)$ and also $F_0(\gb)$) (by taking $A_0(\gb)$ from (2.20)). \enddefinition \remark{Note} Assuming the existence of the cluster expansion for the partition functions in the expression (2.21) for $F_{\text{formal}}$ one sees that the true analogy of the quantity $F_{\text{formal}}$ is $F_{\full }(\gb)$, not $F(\gb)$. However, $F_{\full }(\gb)$ is a {\it nonlocal\/} quantity (though a very quickly converging sum of local quantities $k_T$) and this would make the expansions constructed below in Theorems 3, 4, 5 practically useless. (However, we will return to the value $F_{\full }(\gb)$ in later sections.) On the other hand, $F_{\loc}(\gb)$ is a perfectly local quantity, but sometimes it is ``too crude'' (and therefore never used below) approximation to $F_{\text{formal}}(\gb)$; such a situation happens in the cases where there are {\it very\/} big flat ``ceilings'' on the boundary of $V(\gb)$; a situation having no analogy in the translation invariant situation where the choice of $F_{\loc}(\gb)$ would be without problems because contours are ``crusted'' in that case. The quantity $F(\gb)$ is a reasonable {\it compromise\/} because it approximates \ $F_{\full }(\gb)$ \ with a great accuracy ($\sim \exp(-\tau | \supp \gb|$) and at the same time it is ``local'' in a reasonable sense. One could take even smaller sets $\vv \supset V(\gb)$ to retain this accuracy -- but our choice has an advantage to be in conformity also with previous topological constructions, namely with our approach to the notion of a contour. The quantity $F_{\full,\Cal D , \Lambda }(\gb)$ is just a temporary notation used in the proof of Theorem 3 below. \footnote{ One general ``philosophical'' remark: sometimes, one is fighting severe technical problems in the \ps \ theory which however are relevant only in volumes which are really {\it astronomically large\/}; for example the problem mentioned above (namely the ambiguity in the value of different variants of $A(\gb)$) is hardly of much relevance in volumes of medium sizes like $10^{27}$ !} \endremark The forthcoming theorem is an {\it essential\/} technical step in our procedure of ``recoloring of an internal contour system''. This is further developed by Theorem 4 and finally by Theorem 5 of the next section where we return back to horizontally invariant models. \footnote{Thus, our Theorems 3 and 4 are just some {\it technical steps\/} used merely to prove Theorem 5. We admit that a more direct proof of Theorem 5 could be also given, by formulating Theorems 3 and 4 directly for {\it collections\/} of compatible shifts of $\gb$, thus avoiding (at the price of more complicated formulation of relations like (3.23), (3.36) etc.) the use of horizontally noninvariant mixed models.} \proclaim {Theorem 3} Assume that we have a mixed model satisfying (3.1). Let $\Lambda \subset \zv$. Let and $\gb$ and $\Cal D$ be two disjoint contour systems such that $\Cal D \cup \gb$ is also an admissible contour system. Denote it by $\Cal D\ \&\ \gb$, assume that $\gb$ is ``strictly internal in $\Lambda$'', satisfying the condition $ \vv \cap \cap (\Cal D \cup \Lambda^c) =\emptyset$. Let $\alpha \in \es $ \footnote { It is sufficient to have the condition $\alpha \in \es$ only ``locally'' on $\partial (\Lambda)^c$, and actually we will need this more general case at the very end of our paper, when deriving the relation (3.93).} be a boundary condition on $\partial \Lambda ^c$ which is in conformity with both $\Cal D$ and $\Cal D\ \&\ \gb$ i.e. such that $(x_{\Cal D\&\gb})^{\text{ext}} = (x_{\Cal D})^{\text{ext}}=\alpha$. Then $$ Z_{\Cal D\&\gb}^{\alpha }(\Lambda )= Z_{\Cal D}^{\alpha }(\Lambda ) \exp(-F(\gb)) \exp(\sum k_T^{\text{corr}}) \tag 3.23 $$ where $k_T^{\text{corr}}$ (it is given below by (3.26) and (3.27)) depends on $x_{\Cal D \& \gb}|\vvt$ \footnote{For typhographical reasons, $x_{\gb}|V$ denotes here (again) the restriction $y_V$ of the configuration $y = x_{\gb}$.} and $x_{\Cal D}|\vvt$, $\vvt = \cup_{G \in T} \vvg$, and the summation is over all $T\ssubset \Lambda, T\cap \cap \vv^c \ne \emptyset $ ``touching $\gb$'' such that $\dist(T,\supp\gb) \leq 4 \diam T$. The quantities $k_T^{\text{corr}} $ satisfy for each $T$ the bound $$ |k_T^{\text{corr}}| \leq 2\,\varepsilon ^{\conn T}. \tag 3.24$$ \endproclaim \demo{Proof} Write $\gamma \delta $ resp\. $\delta $ instead of $x_{\Cal D\&\gb}$ resp\. $x_{\Cal D}$. Write $Z_{\Cal D\&\gb}^{\alpha }(\Lambda )$ as $$ \aligned &\exp(-E(\gb)-E(\Cal D )) \exp(-\sum_{t\in\Lambda }e_t(\gamma \delta )) \exp(\sum\Sb T\ssubset\Lambda \setminus (\supp \Cal D \cup \supp \gb)\endSb k_T(\gamma \delta ) )=\\ &=\exp(-E(\Cal D )) \exp(-\sum_{t\in\Lambda }e_t(\delta )) \exp(-F_{\full,\Cal D , \Lambda }(\gb)) \exp(\sum\Sb T\ssubset\Lambda \setminus \supp \Cal D \endSb k_T(\delta ) )=\\ &=Z_{\Cal D}^{\alpha }(\Lambda ) \exp(-F(\gb)) \exp( \sum \Sb T\ssubset \Lambda \\ T\ccap ( \vv)^c \ne \emptyset \\ T \ccap (\supp\Cal D \cup \supp \gb) = \emptyset \endSb k_T(\gamma \delta )- \sum \Sb T\ssubset \Lambda \\ T\ccap( \vv)^c \ne \emptyset \\ T \ccap (\supp\Cal D) =\emptyset \endSb k_T(\delta ) ) \endaligned . \tag 3.25 $$ and this relation \footnote{In the last expression, we just write the detailed expression of $F(\gb) - F_{\full,\Cal D , \Lambda }(\gb)$.} proves (3.23) with the following choice of the quantities $k_T^{\text{corr}}$ (recall that the ``old'' quantities $k_T$ satisfy (3.1), hence we have the bound (3.24)) $$ k_T^{\text{corr}}=k_T(\gamma \delta ) - k_T(\delta ) \tag 3.26$$ if $ T\ssubset \Lambda\, , \, \, T\ccap ( \vv)^c \ne \emptyset\, , \, \, T \ccap (\supp\Cal D \cup \supp\gb) = \emptyset, \vvt \cap V(\Gammab) \neq \emptyset $, resp. $$ k_T^{\text{corr}}=-k_T( \delta ) \tag 3.27 $$ if $ T\ssubset \Lambda\, , \, \, T\ccap( \vv)^c \ne \emptyset \, , \, T \ccap \supp \Cal D = \emptyset \, , \ T \ccap \supp\gb \neq \emptyset $. The last condition $ \vvt \cap V(\Gammab) \neq \emptyset$ in (3.26) is obviously necessary for a {\it nonzero\/} result in (3.26). To check that such a condition implies also the condition $ \dist(T,\supp\gb) \leq 4 \diam T$ (the latter condition we automatically have in (3.27), with the constant 1 instead of 4) combine the condition $\vvt \cup V(\gb) \ne \emptyset$ with the condition $T\ccap ( \vv)^c \ne \emptyset$. Then use the following observation. Consider what the relations stated above for $T$ mean for any element $G \in T$ (warning: $G \ne \supp \gb$): Namely, then (the following argument is for a cluster $T =\{G,\emptyset\}$, the estimate in the general case is even better) the validity of $$ G\cap\cap (\vv)^c \ne \emptyset \ \ \& \ \vvg \cap V(\gb) \neq \emptyset \tag 3.28 $$ would imply (draw a picture) that the integers $d,g,m$ denoting the distance of $G$ from $\supp \gb$ resp. the diameter of $G$ resp. the diameter of $\supp \gb$ would satisfy the following relation: $ 3g > m \ $ . Thus, $d \leq 1/2 m + g \leq 5/2 g < 4g$. \footnote{Notice that any point of $V(\gb)$ has a distance at most $1/2 \ m$ from $\supp \gb$ and hence $d < 1/2 \ m +g$. Also, $\vv = V_{\updownarrow}(V(\gb)$) and the distance of $G$ from $(\vv)^c$ is at most $ g$ if $G \cap \cap (\vv)^c \ne \emptyset $.} \enddemo \definition{Notation} A subsystem $\gb$ of a contour system $\gb'$ satisfying the condition of ``isolation'' $ \supp \gb \ccap (\supp \gb' \setminus \gb) = \emptyset$ will be called a {\it strictly interior\/} contour (sub)system of $\gb'$. \enddefinition \proclaim{Theorem 4} Assume that we have a mixed model $\Cal M$ satisfying (3.1). Consider the partition functions (3.19), with additional summation over $\gb$ (with a fixed $\supp \gb = G$): $$ Z_{\Cal D, G}^{\alpha}(\Lambda) = \sum_{\gb:\supp\gb=G}Z_{\Cal D \&\gb}^{\alpha }(\Lambda) \ \ \ \ \text{and }\ \ \ \ Z_{\Cal D, [\,G\,]}^{\alpha}(\Lambda) = Z_{\Cal D}^{\alpha}(\Lambda) + Z_{\Cal D, G}^{\alpha}(\Lambda) \tag 3.29$$ where $\Cal D \& \gb$ is a contour system such that $\gb$ is its strictly interior contour subsystem. The partition functions above correspond to the events ``\ $\Cal D \ \& \ \gb $ appears'' resp. `` $\gb: \supp \gb = G$ could appear with $\Cal D$''. These partition functions can be expressed correspondingly as $$ Z_{\Cal D, G }^{\alpha }(\Lambda )= (\sum_{T \ssubset \Lambda:G\in T} k_T^+(\delta))Z_{\Cal D}^{\alpha}(\Lambda) , $$ $$ Z_{\Cal D, [G]}^{\alpha}(\Lambda)=\exp(\sum_{T \ssubset \Lambda:G\in T} k_T^*(\delta)) Z_{\Cal D}^{\alpha}(\Lambda) \tag 3.30 $$ respectively; recall that $G\in T$ means that $G$ is the core of the cluster T and $k_T^+(\delta)$ resp. $k_T^*(\delta)$ are some new cluster terms, described in detail in the proof. The leading new quantity $k_G^*(\delta) = k_G^+(\delta)$ is equal to $$ k_G^+ = k_{G}^+ ( \delta) = \sum_{\gb:\supp \gb = G} \exp (-F(\gb)). \tag 3.31 $$ where the sum is over all contours having a given support $G$ and a given external colour $\delta$ determined by $\alpha$ and $\Cal D$. The remaining new quantities $k_T^+$ obey the bounds, for any new cluster of the type $T =(G, \{T_i\})$ (where $T_i \ ;i = 1, \dots, m $ are clusters of $\Cal M$) $$ |k_T^+(\delta)| \leq k_G^+ (\delta)\ 2^m \varepsilon ^{\ \sum_{i=1}^m \conn T_i}\, \tag 3.32 $$ and analogous bounds are valid for $k_T^* $. \footnote{See (3.41) for a more precise estimate on the right hand side; also for more complicated clusters $T$.} The quantities $ k_G^+,\ k_T^+,\ k_T^* \ $ do not already depend on $\gb$; they depend only on $ G$ and on the values $\delta_G$ \footnote{See Theorem 3; $k_G^+$ does not depend on the interior colour of $\gb$ in contrast to $\exp(-F(\gb))$.} and they are translation invariant in the sense of (3.16). Moreover, if $\ k_G^+(\delta) $ satisfies a bound of the type $$k_G^+ (\delta)\leq \varepsilon ^{\Conn G} (\varepsilon')^{\diam G}\tag 3.33 $$ with another small $\varepsilon'$ (this is a simpler variant of (3.8)) then the validity of (3.1) in the given mixed model $\Cal M$ implies its validity \footnote{ Even with some ``reserve'', see (3.41).} also for the new quantities $k_T^+$ and $k_T^*\ $. \footnote{Theorem 5, next section, formulates a ``horizontally invariant'' modification of Theorem 4. The constant $\varepsilon$ is used throughout in bounds like (3.1) and its value is assumed to be sufficiently small, to guarantee Proposition 2, section 1 and therefore the quick convergence of all infinite sums.} \endproclaim \definition{How to use Theorem 4} The procedure called ``recoloring of a contour system'' described by formula (3.30) will be used later repeatedly many times for all the ``smallest possible'' strictly interior contour systems $\gb$. The new cluster quantities $k_T^*$ will play the same role as the ``old'' quantities $k_T$ and therefore it is crucial to ensure (through (3.33)) the validity of (3.1) for them. This means that the new mixed model constructed by the formula (3.30) (and which has a new, richer family of cluster fields $\{k_T\} \& \{k_T^*\}$ but which does not yet allow $\gb$ as a strictly interior contour system of its configuration) is of the {\it same type\/} as before, satisfying again the estimate (3.1). However, first we will replace (3.33) by a more comfortable estimate: \enddefinition \definition{Recolorable contour systems} This notion will be defined (in view of Theorem 5) only for horizontally invariant mixed models. If we have a possibly noninvariant model (like here in Theorems 3, 4) we define the {\it (horizontally) invariant part\/} $\Cal M_{\text{inv}}$ of the model $\Cal M$ by \footnote{We do not mean here, of course, the noninvariancy caused by the fact that we are in a finite $\Lambda$!} discarding all the values $k_T(y)$ which are {\it not\/} horizontally invariant. Now we will say that a strictly interior contour system $\gb$ is {\it recolorable\/} (more precisely $\hat \tau$--recolorable) in the mixed model $\Cal M$ if the inequality ($F_{\text{inv}}$ is taken in the model $\Cal M_{\text{inv}}!$) $$ \sum^{x_{\gb}^{\text{ext}}= y}_{\gb :\supp \gb = G}\exp (-F_{\text{inv}}(\gb)) \leq \exp(-\hat \tau \conn G) \tag 3.34 $$ holds with a large $\hat \tau$, for each possible external colour $y = x_{\gb}^{\text{ext}}$ given on on $\vv$. \enddefinition \remark{Notes} 1. Notice that this is the requirement on the {\it set \/} $G$ and the external colour $y$, not on a particular contour system $\gb$ -- though practically this is closely related to saying that for each $\gb$ with the same support $G$, one has a bound, with some other large $\hat \tau^*$, $$ F_{\text{inv}}(\gb) \geq \hat \tau^* \conn \gb . \tag 3.34*$$ 2. One should always have in mind that the property ``to be recolorable'' is rather sensitive (for contour systems of {\it very\/} large size, of course) to the boundary conditions. The mere knowledge of the external colour of $\gb$ only ``at the vertical level of $V(\gb)$'' may be insufficient to decide the recolorability. In general, the external colour of the whole volume $\vv$ (some logarithmic neighborhood of $ V(\gb)$ would suffice, in fact) must be known. 3. In practice, the requirement (3.33) can be checked through the recolorability (3.34) of $\gb$, with $\hat \tau $ satisfying (with some ``reserve'') the inequality (compare (3.31) and (3.33)) $$ \exp(-\hat \tau\conn G) < \varepsilon ^{\Conn G} (\varepsilon')^{\diam G}. \tag 3.35 $$ Then the relation (3.35) implies also (3.33). Namely, only ``slightly noninvariant'' models, with $F(\gb) - F_{\text{inv}}(\gb)$ being of the order $\varepsilon^{\diam \gb}$ only, appear when applying Theorem 4 successively (by recoloring all the shifts of some $\gb$, see the proof of Theorem 5) to some invariant mixed model. 4. Thus, the bound (3.35) relates the appropriate choice of the constants $\varepsilon, \hat \tau$\ in (3.1) and (3.34). The latter condition is a {\it Peierls type condition\/} for the quantity $F(\gb)$. The possible choice of $\hat \tau = c \tau $ (such that (3.34) could be checked; $\tau$ is from (2.14)) will be discussed below (see (3.59); the convenient choice of the constant being $c= 1 / 12\nu $). \footnote{It is not at all necessary that $\hat \tau$ in (3.34) would be absolutely the same for all $\gb$. For different shapes of $\gb$ one can take different (large) $\hat \tau$. However, it is technically important (for easy formulation of Theorem 5) that recolorability would hold for all the horizontal translates of $\gb$ at once.} \endremark 5. One should also emphasize that when performing later our successive process of ``recoloring of all (recolorable) $\gb$'', our procedure will be organized in such a way (see the forthcoming section) that quantities $k_T^*$ with {\it new\/} clusters $T$ (nonexistent with nonzero $k_T$ in the previous mixed model) will appear at each stage of the construction. \endremark \demo{Proof of Theorem 4 } The proof is based on Theorem 3 and Proposition 2, section 1: By (3.23) we have the relation $$ Z_{\Cal D, G }^{\alpha }(\Lambda)= \sum_{\gb:\supp\gb=G} Z_{\Cal D \&\gb}^{\alpha }(\Lambda)= Z_{\Cal D }^{\alpha }(\Lambda) \sum_{\gb} \exp(-F(\gb)) \exp(\sum_T k_T^{\text{corr}})\, .$$ Writing $\exp(-F(\gb)) = \xi _{\gb} k_G^+$ and expanding the exponential this can be written as $$ Z_{\Cal D, G }^{\alpha }(\Lambda) = Z_{\Cal D }^{\alpha }(\Lambda) \biggl(\sum_{\gb} \xi _{\gb} \sum_{k=0}^{\infty} \sum\Sb (T_1, \dots, T_k)\\ (n_1, \dots, n_k) \endSb \prod_{i=1}^{k}\frac{1}{n_i!}(k_{T_i}^{\text{corr}})^{n_i})\biggr) k_G^+ = Z_{\Cal D }^{\alpha }( \Lambda) \ \sum_T k_T^+ .\tag 3.36$$ The summation is over new clusters T (see below) satisfying analogous properties as in (3.23) and the new values of $k_T$ are defined here as \footnote{Notice that whereas $F(\gb)$ depends on the values $x_{\gb}| \vv$ , in particular on the ``interior colour of $\gb$'', the quantity $k^+_T$ depends only on $G$ and the ``external colour'' of $\gb$ restricted to $\vv$.} $$ k_T^+ = \sum\Sb \gb:\supp \gb = G\endSb \xi _{\gb} k^+_G \prod_{i=1}^{k}\sum_{n_i \geq 1} \frac{1}{n_i!}(k_{T_i}^{\text{corr}})^{n_i} . \tag 3.37$$ The clusters $T$ are defined (for $n_i=1$; notice that there is a summation over $n_i \geq 1$) as $$ T= (G, \{T_i\}) . \tag 3.38 $$ the most important term \footnote{ This is the point where our hierarchical construction of clusters is rather important, to assure that we really have {\it new\/} clusters here, which were not used before (e.g. in (3.23)). Not having this, the condition (3.1) could have been spoiled after adding several new values of $k_T$ for the same cluster $T$ !} corresponding to the empty collection of $\{T_i\}$. Notice that the expression of $k_T^+$ by values $k_T^{\text{corr}}$ is not exactly as (3.7), Proposition 1 of Section 1. However, it is straightforward to adapt the corresponding estimates noticing that $\sum_{\gb} \xi _{\gb} = 1$. Thus, one obtains (3.32) (the term $2^m$ is from (3.24) !) and then, from (3.34) and (3.35), also (3.1) for the quantity $k_T^+$. Namely the term (emerging, for the new cluster $T = \{G,\{T_i\}\}$, if we directly substitute (3.32) and the bound (3.33) for $k_G^+$ into (3.37)) $$ 2^n \ \varepsilon^{ \sum_{i=1}^n \conn T_i + \conn G } \ (\varepsilon')^{\diam G} $$ is surely smaller than $\varepsilon^{\conn T}$ as some inspection of the notion of $\conn T_i$ shows. The expansion (3.31) is now obtained by taking logarithms. We get \ \ $\log Z_{\Cal D, [G]}^{\alpha}(\Lambda) = $ $$ \log Z_{\Cal D}^{\alpha}(\Lambda)+ \log(1+ \sum_{(T_1,T_2,\dots,T_n) \ssubset \Lambda}^{ (T_1,T_2,\dots,T_n)\owns G} k_{(T_1,T_2,\dots,T_n)}^+) = \log Z_{\Cal D }^{\alpha }(\Lambda)+ \sum_{(T_1,T_2,\dots,T_n) \ssubset \Lambda}^{(T_1,T_2,\dots,T_n)\owns G} k_{(T_1,T_2,\dots,T_n)}^{(*)} \tag 3.39 $$ where \footnote{ For typhographic reasons, we write part of the requirements on the summands above the sum.} (notice that {\it all\/} the clusters $T_i$ mentioned in these relations contain the set $G$) $$ k_{(T_1,T_2,\dots,T_n)}^{(*)}= \frac{(-1)^{n-1}} {n} \prod_{i=1}^{n}k_{T_i}^+ \tag 3.40$$ for any ``cluster'' $(T_1, T_2, \dots, T_n) $; and one has to modify correspondingly this formula if multiple copies of one cluster $T_i$ appear in $T$. However, we will {\it not\/} define ``clusters'' of a type $(T_1,T_2,\dots,T_n)$. Instead, we perform a {\it partial summation\/} over all these collections $(T_1, T_2, \dots, T_n) $ with the same $ T =\cup \ T_i$ in the relation \footnote{By $\cup \ T_i$ we mean the cluster $(G,\{T^j_i\})$ where $T_i =(G,\{T^j_i\})$; each $T^j_i$ being counted only once in $T$.} (3.39), namely we put $$ k_T^* = \sum_{ (T_1,T_2,\dots,T_n)} k_{(T_1,T_2,\dots,T_n)}^{(*)} \tag 3.40' $$ where the summation is over all $(T_1, T_2, \dots, T_n) $ with the same union $T =\cup \ T_i$ . Now, for any such $n$ -- tuple $(T_1, T_2, \dots, T_n)$ (recall that $G\in T_i$ for each $ i$) one obtains, after some inspection, the bound $$ |k_{(T_1,T_2,\dots,T_n)}^{(*)}| \leq (k_G^+)^n\ \varepsilon ^{\sum_{i=1}^n \conn ^+ (T_i)} \ \ \ \ \ \text{i.e.} \ \ \ \ \ k_T^*\leq \varepsilon ^{\conn T}\tag 3.41$$ where $\conn^+ ( T) $ denotes the quantity $ \sum_j \conn T_j$ for $T=(G, \{T_j\})$; the last inequality follows from (3.34) and (3.35) by a similar argument as we used in the estimate of the values (3.37) above. This concludes the proof of Theorem 4. \enddemo \vskip1mm \head 4. Recoloring: towards a new stratified mixed model \endhead The aim of this section is to formulate a procedure (based on Theorem 4) which converts a given {\it horizontally translation invariant\/} mixed model into a new horizontally translation invariant model, having the same ``diluted ''partition functions but with a {\it smaller\/} set of allowed configurations (and with a {\it richer\/} set of clusters $T$ having nonzero contributions $k_T$; the ``old'' nonzero values $k_T$ being kept at the {\it same value\/} as before). Such a transformation of the model could be characterized as the ``removal, from the model, of all configurations which have a shift of $\gb$ among its strictly interior contour systems''. Recall the ordering $\prec$ (and $\prec \prec$) of contour systems we introduced in section III,1. We will say that a recolorable contour system $\gb$ is {\it smallest recolorable\/} contour system of the given mixed model if it is strictly interior in the considered volume $\Lambda$, recolorable and moreover there is no smaller, in the ordering $\prec$ (resp. $\prec \prec$ if we work with horizontally invariant models) , strictly interior recolorable $\gb'$ which could appear in some configuration of the given mixed model. \remark {Note} Below we will use the recoloring step formulated by Theorem 5 {\it successively\/}, according to the growing ``size'' of the smallest recolorable contour systems $\gb$ which have to be recolored. Moreover, the mixed models studied by us later will appear as the result of successive recolorings applied to some given \ps \ abstract model; the configuration spaces of the mixed models thus obtained will be defined in terms of requirements on the {\it size\/} of the smallest interior recolorable contour systems of the configuration. See the forthcoming sections for details. \endremark \definition{The notion of an equivalent mixed model} Recall that we say that a cluster $T$ {\it contains\/} a contour system \footnote{Of course this is a slight abuse of notations because clusters are made of {\it sets\/} $\supp \gb$, not of contours $\gb$. However, we always keep in mind the {\it external colour\/} $y$ of $\gb$.} $\gb$ if the set $\supp \gb$ was used, with the external colour $x_{\gb}^{\text{ext}}$, as a core of some intermediate cluster in the recursive construction of $T$. Two mixed models will be said to be equivalent if all their strictly diluted partition functions (for all finite volumes and boundary conditions $y \in \es$) are the {\it same\/}. We notice that usually, the configuration space of one of these two mixed models will be a suitable {\it subset\/} of the configuration space of the other model; on the contrary the collection of clusters with nonzero $k_T$ for the model with a smaller configuration space will be usually {\it richer\/}, including also some clusters containing contours which are present only in the model with a bigger configurations space. This is the case of the following result.\enddefinition \proclaim{Theorem 5} Assume that we have a horizontally translation invariant mixed model satisfying the condition (3.1). Consider any finite volume $\Lambda$. \footnote{Theorem 5 is applied in all finite volumes {\it at once\/}, for any fixed $\supp \gb$ and $y_{\vv}$.} Let $\gb$ be a recolorable \footnote{ In particular, we may assume that $\supp \gb$ is smallest possible in the relation $\prec$; \ this will be the case needed in the applications of Theorem 5 below. We assume that $\hat \tau$ from (3.34) is sufficiently large !} contour system whose horizontal shift can appear as a strictly interior contour system of some configuration of the model. Denote by $y$ its external colour. Assume that $k_T(y) = 0 $ holds for all clusters $T$ containing a shift of $\supp \gb$. Then there is an equivalent mixed model having the following properties : \roster \item Its configuration space is the collection of all configurations of the original mixed model which do not contain a horizontal shift of some $\gb'$, $\supp \gb =\supp \gb'$ with the same external colour $y$, among its possible strictly interior recolorable subsystems. \item If $T$ is a cluster not containing a horizontal shift of $\supp \gb$ then the value of $k_T(y)$ in the new mixed model is the same as before. \item If $T$ contains a horizontal shift of $\supp \gb$ then the new value of $k_T$ satisfies the condition (3.1), too, assuming that $\varepsilon$ and $\hat \tau$ are such that (3.34) holds. \endroster \endproclaim \remark{Note} Notice that when applying Theorem 5 to one particular $\gb$ it does not mean that some other $\gb'$ with the same support $\supp \gb' =\supp \gb$ with the external colour $y'$ differing from $y$ on $\vv$ would not still {\it remain\/} in the resulting mixed model. So we have to use Theorem 5 {\it repeatedly\/} again -- to remove all the recolorable systems $\gb$ with the same support $\supp \gb$ but with different external colours $y$. The recoloring of various contour systems $\gb$ with different externals colour $y$ and the same $\supp \gb$ is made independently, the corresponding expansions not affecting each other! \endremark \demo{Proof} This follows from Theorem 4 if we use it successively in the following way: take the (completely ordered in $\prec$ ) sequence of all shifts of $\gb$. Given $t \in \zv$ consider an intermediate ``$t$--th model'' \footnote{This has sense in any {\it finite\/} volume $\Lambda$, and the cluster terms $k_T^*$ do {\it not\/} depend on the volume; however the summation over $T$ depends on $\Lambda$.} defined below which has the configuration space defined by the requirement that exactly those configurations of the original mixed model are allowed for which {\it no\/} strictly interior contour system $ \gb + t'$ such that $ t' \prec_l t $ exist. If $s$ is the nearest greater point to $t$ ($ t \prec s $) in the given finite volume $\Lambda$ then we define the transition to the $s$--th model by the very procedure described by Theorem 4. It is straightforward to check the translation invariance (3.16) and uniqueness of the definition (not depending on the actual volume $\Lambda$; recall that $T \ssubset \Lambda$) of all the new quantities $k_T$ thus obtained. \footnote{ There are modifications possible in the method we used to prove Theorem 5. Namely, it is possible (and it is, in fact, apparently a more standard way how to deal with these cluster expansions) to reformulate the Theorem 3 and Theorem 4 above into a single statement describing {\it simultaneous recoloring\/} of all the shifts of $\gb$ {\it at once\/}. We do not follow such a (more direct, but with slightly more complicated formulas) approach in this paper. However, such an approach is used also in the recent lectures \cite{ZRO} (in a simplest possible form, we believe) and in future publications, we plan to replace the arguments based on the successsive use of the lexicographic order by this more standard approach.} \enddemo \head 5. Small and extremally small contour systems \endhead We are still working with a {\it general mixed model\/}. Only later we will explain the relevant choice of a mixed model in a concrete situation; this choice will be always given as a suitable partial expansion of the original abstract Pirogov -- Sinai model. The successive application of the recoloring procedure constructed in the preceding chapters (culminating in Theorem 5) will finally lead, in any volume, to a family (indexed by elements of $\es$) of mixed models where {\it no\/} recolorable systems will be left! The reader is advised to look briefly to the section 8 and to look at the formulation and the proof of Main Theorem -- to see the important consequences of this fact. In the meantime, in sections 5 to 7, we will investigate the notion of a recolorable contour system (more precisely the related notion of a small resp. extremally small contour system introduced below) in a more depth. We will construct an important concept of a ``skeleton'' of a contour system $\gb$; this notion will be investigated using some useful supplementary ``topological'' results derived later, namely Theorems 6 and 7. All this will be needed in the proof of the Main Theorem. Possibly, our discussion of the notions of a ``small'' and of an ``extremally small'' system, and our definition of the notion of a ``skeleton'' of $\gb$ (notice that we will introduce there another testing quantity $A^*(\square)$, as a more careful alternative to $A(\gb) $) is slightly more detailed than it is absolutely necessary -- if a shortest possible proof of Main Theorem is required. However, we are keeping all these seemingly unnecessary details here -- also because we expect our more detailed exposition will be useful not only here (giving more information and estimates with better constants compared to a more crude method) but also in the future investigations of the ``metastability'' problem and in the study of the completeness of the phase picture constructed by our Main Theorem. The reader interested in acquiring the {\it idea of the proof\/} of Main Theorem can go now directly to Section 8, omitting even the very notion of a small contour system but finally realizing that some variant of such a notion (and of Theorem 7) is needed there! \definition{ Preparation: Relations between different variants of $A(\gb)$} \enddefinition Recall the definition of $A(\gb)$, $A_{\text{full}}(\gb)$, and of the corresponding quantities $F(\gb)$ from Section 3. When checking the condition (3.34) or (3.34*) one needs (at least for contour systems whose energy is not much above the bound assumed in (2.14)) an inequality $$ A(\gb) \leq \tilde \tau |\supp \gb | $$ where $\tilde \tau < \tau$ is a suitable constant ``not too close to $\tau$'' ($\tau$ from (2.14)). \footnote{In fact, there is quite a freedom in the choice of all these constants $\tilde \tau,\ \hat \tau, \hat \tau^*$ from (3.34,34*)) resp. from the relation (3.49) below and the difference between various choices of $A(\gb)$ (and, accordingly of $F(\gb)$) which was so important in Section 3 will be quite irrelevant here, except of the choice of $A_{\text {loc}}(\gb)$ which would be too rough in situations where some {\it really big\/} ``ceilings'' of $\gb$ appear.} In these estimates, it will be more convenient to work with quantities of the type $A_{\text{full}}$ instead of $A$. Fortunately, both quantities are practically the same for any $\gb$. More precisely we have the following bound which is easily obtained if we notice that the cluster terms $k_T$ acting in the difference below -- compare (3.21) -- decay exponentially quickly with the distance of $T$ and $\supp \gb$ (for such an observation it is important to recall that the set $\vv$ -- and not $V(\gb)$, for example-- was used in the definition of $A(\gb)$!). \proclaim{Proposition 1} For any contour system $\gb$ (and for any extension of the boundary condition $x_{\gb}^{\text{ext}}$ to the whole $\zv$), the quantities $A(\gb)$ and $ A_{\text{full}}(\gb)$ satisfy the bound $$ | A(\gb) - A_{\text{full}}(\gb)| \leq (\varepsilon')^{\diam \gb} |\supp \gb| \tag 3.42 $$ where $\varepsilon' = C \varepsilon$ and $\varepsilon$ is from (3.1). \endproclaim The quantities $ A_{\text {full}}(\gb) $ (and, even more importantly, the modified quantities $\Am(\square)$ introduced below) will be used instead of $A(\gb)$ in these final sections. All the bounds of the type $ A_{\text{full}}(\gb) \leq \tilde \tau |\supp \gb | $ will be studied rather for them instead of the original quantities $A(\gb)$. Then we supplement these bounds by (3.42). \footnote{Once again: the quantities $A(\gb)$ remain, of course, to be used in (3.22) but their test ``whether they are dangerously big'' will be done through the closely related but ``more nicely looking'' quantities $A_{\text {full}}(\gb)$. } The meaning of the quantities $A(\Gammab)$ is that they give some information about the ``volume gain of the free energy'' caused by the fact that inside $\Gammab$, possibly some ``more stable'' regime is established. One could ask this question in a more precise way: whether the regime which resides inside $\Gammab$ is the ``best'' possible one and also what is the ``energetically optimal realization'' of such a contour system. Fortunately, one does not need to investigate these questions in more detail, in particular the question ``what is the optimal shape of a contour'' has not to be approached here. On the other hand, the question ``what is the best possible stratified regime to be found inside $\gb$'' {\it will\/} be important in the investigations below and we will approach it as follows: We rewrite, from now on, the quantity $A_{\text{full}}(\gb)$ in a more concise way, replacing the sum over {\it cluster\/} quantities $k_T$ by a more nicely looking (and more flexible) sum of suitable {\it point\/} quantities. These latter quantities are however nonlocal (but very quickly converging limits of local quantities). Introduce the following notations. \definition{Free energy densities of the mixed model} For any mixed model and any stratified configuration $y$ define the quantity $$ f_t(y) =e_t(y)- \sum_{T: t \in T} \frac{k_T(y)}{|T|}. \tag 3.43 $$ For an arbitrary nonstratified $x$, define $f_t(x) =f_t(x_t^{\text{hor}})$ where $x_t^{\text{hor}}$ (we denoted it also as $\hat x_t$) is the horizontally invariant extension of the configuration $x_{\{t_1,\dots,t_{\nu -1} ,(.)\}}$. \enddefinition \remark{Note} These quantities will be very important in the sequel. However, in spite of their ``physical'' meaning which we discover below (they will be interpreted as the ``density, at $t$, of the free energy of the metastable state constructed around $y$'') there is still some arbitrariness in their definition: For example, the modified quantity $$\tilde f_t(y) = e_t(y) - \sum_{t\in T} k_T(y) \tag 3.44 $$ where the sum is over all clusters $T$ such that $t$ is the first point of $\supp T$ in the lexicographic order could be used in the same way. \footnote{ However, the physically important quantities like $\sum_t (f_t(z) -f_t(y))$ are the {\it same\/} for both alternatives, whenever $y$ and $z$ are stratified and differing only on some layer of a finite width.} \endremark \remark{Agreement} Here and below we need to work with configurations $y \in \es$ defined on the whole lattice $\zv$. Let us make an agreement that whenever we have a stratified configuration defined, at the moment, only in a partial way (typical situation: the external colour of some configuration defined in some finite volume) then we extend it \footnote{ In fact, this is a comparable act of arbitrariness like that we used in our choice of the sets $\vv$.} in some way to a configuration on the whole $\zv$. The details of the extension will be usually irrelevant. \endremark \proclaim{Proposition 2} For any contour system $\gb$ and any extension of its external colour $x_{\gb}^{\text{ext}}$, the quantity $A_{\text{full}}(\gb)$ can be expressed by the following formula where $\varepsilon''= C \varepsilon^n$, $n$ being the cardinality of a smallest possible cluster appearing in (3.21) \footnote{ The quantity $\varepsilon $ is from (3.1) and $C =C(\nu)$ is a suitable constant.}: $$ A_{\text{full}}(\gb) = A^*_{\text{full}}(\gb) +\Delta(\gb) \ \ \ \ \ \ \text{where} \tag 3.45 $$ $$ A^*_{\text{full}}(\gb) =\sum_{t \in \zv} (f_t(x_{\gb}^{\text{ext}}) - f_t(x_{\gb})) \ \ \ \ \ \ \text{and} \ \ \ \ \ \ |\Delta (\gb)| \leq \varepsilon'' |\supp \gb| . $$ \endproclaim \demo{Proof} This follows from the observation that any $k_T(x_{\gb})$, analogously $k_T(y), y = x_{\gb}^{\text{ext}}$, $T \ssubset (\supp \gb)^c$ which is counted in $A_{\text{full}}(\gb)$ is counted also (exactly once!) in $$\sum_{t \in \zv} (f_t(x_{\gb}^{\text{ext}})- f_t(x_{\gb})) = \sum_{t \in \zv} (e_t(x_{\gb}^{\text{ext}})- e_t(x_{\gb})) + \sum_t \sum_{T\owns t} \frac {(k_T(x_{\gb}^{\text{ext}})^{\text{hor}} - k_T(x_{\gb}))}{|T|}.$$ The correction terms $k_T |T|^{-1}$) contributing to $\Delta$ thus arise only from clusters satisfying the relation $T\cap \cap \supp \gb \ne \emptyset$ and when summed they give the bound (3.45). Namely $A^*(\gb) - A(\gb)$ can be written, when inserting (3.43) into it and rearranging the terms, as a suitable sum of $k_T(y)/|T|$ over clusters $T$ whose distance to $\gb$ is comparable to their diameter. \footnote{ This observation is immediate for the ``clusters acting already in the sum for $A_{\text{full}}(\gb)$''. The remaining clusters (appearing only in the sums for $f_t$) must satisfy the conditions $T \cap \cap \gb \ne \emptyset $ and also $\vvt \cap V(\gb) \ne \emptyset$ -- to give a nonzero contribution $k_T(\hat x) - k_T(\hat x^{\text{ext}})$. We noted in an analogous situation of Theorem 3 that such clusters $T$ really have a diameter comparable to their distance to $\gb$.} The contribution of these terms is like $C \varepsilon |\supp \gb|$. \enddemo The forthcoming notion will be useful for understanding what would be the ``best possible gain of free energy'' inside a given contour $\gb$ : \definition{Configurations minimizing $f_t(x)$} Given a configuration $x$ which is stratified outside of some finite volume $V$ denote by $x_V^{\text{best}}$ the configuration minimizing, at each vertical section $(t_1,\dots,t_{\nu -1},(.))$, the value $\sum_{t_{\nu}\in \zet} f_t(x')$ under the condition that $x' =x$ outside of $V$. \enddefinition \remark{Note} This notion is close (but not identical) to that of $x^{\text{best}}_{\gb}$ used above. Namely one can consider a configuration $x^{\text{best}}_V$ in a volume $V =V(\gb)$, where $\gb$ is a contour system. However, we do not fix neither the values of $\partial_r \gb$ nor $x_{V(\gb ) \setminus \supp \gb}$ here. We will use the notation $x^{\text{best}}_V$ mostly for {\it cubes\/} $V$. Notice also that in comparison to the formulation of the Peierls condition in Part II, we use the quantity $f_t(x')$ instead of $e_t(x')$ in the definition of $x_V^{\text{best}}$. However, $f_t$ is roughly equal to $e_t$ and the sum of the terms $(f_t -e_t)$ over $\supp \gb$ is again (like in (3.42)) of the order $\varepsilon' |\supp \gb| $. If a contour $\gb$ is such that $\supp \gb = V$ and the outside colour of $\gb$ is everywhere the same stratified configuration $y \in \es$ then the difference between our old value of $x^{\text{best}}_{\gb}$ and the new one $x^{\text{best}}_V$ is a quantity really negligible when checking the validity of Peierls condition. \endremark Using the quantities $f_t$, we now want to define an alternative (with the same intuitive meaning) to $A(\gb)$ which will be more flexible in the fortcoming estimates. \definition{The quantity $A^*(V,y)$} Given any finite volume \footnote{We will use later this quantity not only for volumes $V= \vv$ but also (and mainly) for {\it cubes\/} $V$.} $V$ and any $y \in \es$ \ introduce the quantity $A^*(V,y)$, more precisely $A^*_{\text{max}}(V,y)$ : $$ A^*(V,y) \equiv A^*_{\text{max}}(V,y) = \max\{ \sum_{t \in \zv} (f_t(y) -f_t(z))\} =\sum_{t \in \zv} (f_t(y)-f_t(y_{V}^{\text{best}})) \tag 3.47$$ where the maximum is taken over all $z$ which are equal to $y$ outside of $V $. \enddefinition Compare now (3.47) with (3.45), for $V\supset \vv$: By Proposition 1 and 2 we have $$ A(\gb) - A^*_{\text{max}}(V,y) \leq A(\gb) -A^*(\gb) \leq \varepsilon''' |\supp \gb| \tag 3.48$$ where $\varepsilon''' = C \varepsilon$ and $y$ is (any extension of) the external colour of $\gb$. \remark{Notes} 1. Our preference of this notion (to the previous alternative $A_{\text{full}}(\gb)$ which was defined with the help of summation over {\it clusters\/} $k_T$ is mainly an aesthetical one. Namely, the sums of {\it point\/} quantities will be more convenient in later estimates. Notice that now we do {\it not\/} require any stratification of $z$ (and of $y_{V}^{\text{best}}$) inside $V$. Later (see the section Skeleton) we will introduce, for technical reasons, some new, artificial ``contours'' of the model. These new contours will have the shape of a (large) cube $\square$ living inside of some stratified regime of the actual ``physical'' configuration; their ``energy'' will be defined (it will be of the order $\diam \square$ only) just to compensate the ``volume gain inside $\square$\,'' (arising from the fact that the configuration inside of such an artificial contour will be assumed to ``jump freely into some better regime inside $\square$\,''. The interplay between these formal notions and a comparison to the behaviour of the real contours of the model can be best studied in the language of the quantities $\Af$ . \newline 2. Recall that the quantity $\Af$ requires the knowledge of $y$ in the whole $\zv$. Thus, there is some small arbitrariness in the definition of $\Af(V,y)$ because our $y$ is usually given only in some finite volume. This arbitrariness is compensated by the more transparent form of the right hand side of (3.47) compared to (3.21). \endremark Thus, when estimating the size of the quantities $A(\gb)$ we will work, from now on, with the more convenient quantities $A_{\text{full}}(\gb)$ or even $\Am(V(\gb))$. In fact, it turns out that it is reasonable \footnote{They are some technical subtleties in this recommendation. They will be more clear later, after defining the notion of an extremally small system, in the proof of Theorem 6. See (3.55).} to restrict the whole discussion of the ``dangerously big values of $A(\gb)$'' to the {\it superordinated cubes only\/}: \endremark \definition { Small cubes of $y \in \es$} 1) A cube $\square$ will be called {\it small\/}, more precisely $\tau'$ -- small, with respect to a configuration $y \in \es$ if the following inequality \footnote{Do not care about the particular choice of the constant $\tau' \approx \tau$ here. {\it Any\/} sufficiently big constant would do the job. On the other hand, the advisibility of our very choice of $\diam \gb$ will be clear only later. We mention that the choice of $\partial \square$ instead of $\diam \gb$ here (such an alternative could maybe look more natural as the quantities $|\supp \gb|$ appear otherwise everywhere whenever the energy of a contour system is considered) would cause technical difficulties later.} holds : $$ \Am(\square,y) \leq \tau'\diam \square \tag 3.49 $$ where $\tau'$ is suitably (slightly) smaller than $\tau$, see below. 2) In finite volumes $\Lambda$, there is some ambiguity in the definition above -- depending on how big a neighborhood $\Lambda \setminus \square$ of $\square$ is given with a uniqely given value of $y \in \es$ . (Obviously, a distance of a size $\dist(\square,\Lambda^c) \approx \log \diam \square$ is sufficient to give a precision $\ll \diam \square$ in the value of $\Am(\square,y)$ above.) In the following we will call more precisely $\square$ a small cube of a configuration $y_{\Lambda}$, $y \in \es$ if (3.49) holds for {\it any\/} \footnote{ The usage of this notion will be practically the same as if we would put ``for {\it some\/} extension of $y_{\Lambda}$''. We will work both with the implications of (3.49) as well as with the impplications of its negation, so there is no logical preference between ``some'' or ``all'' in the definition above. Fortunately, this is not of much importance because if we take a sufficiently (slightly !) smaller cube $\square' \subset \square$ then the value of $A^*(\square', y)$ is practically independent on the extension of $y$ outside of $\square$, as we already noted above.} stratified extension $y$ of $y_{\Lambda}$ to $(\square)^c$. \enddefinition \definition {The choice of $\tau'$} In the following we {\it fix\/} a suitable (slightly smaller than $\tau$) $\tau' < \tau$ guaranteeing the bound $ \tau' < \tau - \varepsilon' - \varepsilon'' -\varepsilon'''$; compare (3.42), (3.45), and (3.48). \footnote{Notice that while in the definition of smallness we fix the choice of a constant $\tau' \doteq \tau$, much smaller constant $\hat \tau \doteq \tau/12 \nu$ has to be used, for technical reasons, in the definition of recolorability.} \enddefinition Say that a rectangle $\square$ in $\zv$ is the {\it covering rectangle\/} of a volume $S \subset \zv $ if \ $ \square$ is the smallest possible rectangle (smallest in the usual partial ordering $\prec$ determined by the diameters of the considered sets) which is a superset of $S$. The {\it cube\/} having the same center as the covering rectangle and also the same diameter (recall that we have the $l_{\infty}$ norm everywhere, so this cube contains the above rectangle as its subset) will be called the {\it covering cube\/} of $S$. We will denote the covering {\it cube\/} (rectangles would be inconvenient to work with, and therefore they are abandoned below) $\square$ by a symbol $ \square(S)$. Instead of $\square(\vv)$ we will also use the notation $ \square(\gb)$. \definition {Small contour systems} If $\square$ is the covering cube of $\gb$ and $y$ is the external colour of $\gb$ then we will say that $\gb$ is {\it small\/} if $\square$ is small with respect to $y$. More precisely we will restrict this notion only to {\it strictly interior contour subsystems\/} $\gb$ of a given admissible system $\Cal D$ in $\Lambda$, and we will say that $\gb$ is {\it small in\/} $\Lambda$ if the covering cube $\square(\gb)$ is small for each stratified extension of the exterior colour of $\gb$ induced by $\Cal D$ and $y$. \enddefinition \remark{Note} The property ``to be small'' will {\it not\/} be formulated for contour systems $\gb \sqcap \Lambda^c \ne \emptyset$. It is easy to see (it is just the monotonicity of $A^*$ with respect to a growing volume) that for any small $\gb$ we have the inequality, with $y$ denoting the external colour of $\gb$ (extended to the whole $\zv$, otherwise some small corrections have to be added) $$ A^*_{\text{max}}(V(\gb),y) \leq A^*_{\text {max}}(\vv,y) \leq A^*_{\text{max}}(\square(\gb),y) \leq \tau' \diam \gb. \tag 3.50 $$ Complement this with (3.42) and (3.45)! The idea now is, roughly speaking, that all small contour systems should be {\it recolorable\/}. This is obviously true for contour systems consisting of {\it one contour only\/} because then we have from (3.50) and (3.42) the following inequality (see (2.15)): $$ F(\gb) = E(\gb) - A(\gb) \geq (\tau^* -\varepsilon)|\supp \gb| - \tau' \diam \gb \geq { \tau^* \over 2} |\supp \gb| \tag 3.51 $$ and the last term is bigger than, say $ \tau^* /12\nu \ \conn \gb$ because of Theorem 7 below. However, we will often need to ``recolor'' also some {\it more complicated\/} strictly interior contour systems i.e. systems of precontours $\gb$ with unclear relation between $|\supp \gb|$ and $\conn \gb$. In such a case, the corresponding more general argument valid for any contour system i.e. for any admissible system of precontours $\gb$ will be developed in (3.54) below. However, the notion of ``smallness'' has to be sharpened there and the arguments (3.51) should be replaced by some more detailed bounds given below, in (3.60). \definition{ Extremally small contour systems} We will say that a small, strictly interior contour system $\gb$ of a configuration $(x, \Cal D)$ is {\it extremally small\/} (more precisely $\tau'$--extremally small) in $\Cal D$ if it is small and moreover if the following requirement for $\gb$ is satisfied: for {\it no\/} strictly interior contour system $\gb' \subsetneqq \gb$, $\gb'$ is extremally small. \footnote{One could substitute here ``small'' or even ``recolorable'' $\gb'$ (instead of extremally small $\gb'$) without changing the sense of the definition. However, a modification of such a notion called ``extremal recolorability'' would be quite impractical. We {\it will need\/} to know that extremal smallness implies recolorability!} \enddefinition \remark{Notes} {\bf 0.} A contour system $\gb$ will be later called {\it removable\/} if it is either extremally small or (more generally, see below) recolorable and it is moreover the smallest one, in the ordering $\prec \prec $ among such contour subsystems of a given mixed model. We may inform the reader in advance that below we prove that the extremal smallness of $\gb$ implies its {\it recolorability\/}, and therefore also its removability in the case of a smallest possible size of $\gb$! However, smallness is a much nicer property to work with; to check it suffices to look only on the covering cube $\square(\gb)$ of $\gb$ and to estimate the quantity $A^*(\square)$, which is a more pleasant task than to estimate $F(\gb)$. The {\it minimality of size\/} in the definition above just reflects our desire to organize the recoloring by suitable {\it induction over the size of $\gb$\/}. \newline {\bf 1.} We can claim now that any contour system $\Cal D$ has the following property: after the removal of a removable contour system $\gb$ (more precisely, of all the shifts $\gb' \sqsubset \Lambda$ of $\gb$ according to Theorem 5) from $\Cal D$, no removable contour system $\gb'$ of $\Cal D$ remains such that $\gb' \prec \gb$, and the same remains true if subsequent removals are applied. This property will be rather convenient \footnote{ It will cause all the newly defined clusters to be the {\it standard\/} ones.} later when applying successively the procedure of recoloring, by Theorem 5, to a given model. \newline {\bf 2.} The adjective ``small'' resp.``extremally small'' has only a loose relation to the actual {\it size} of $\supp\gb$. It is in accordance with the usage of this term (and also of the related, perhaps confusingly sounding term ``stable'') in [Z]. There are other adjectives used to describe such a property in the literature -- like ``damped'' in [K]. \newline {\bf 3.} A typical example of an extremally small system $\Cal D $ is a collection of the type $\Cal D =\gb_{\ext}\ \&\ \{\gb_i \}$ where $\gb_{ext}$ is an ``external'' contour system (one has to be careful with such a notion, see below), where the ``internal''contour systems $\gb_i$ are {\it not\/} small but the whole system $\Cal D $ {\it is\/} small. The case $\{\gb_i\}=\emptyset$ is the most important and common one, of course. \newline {\bf 4.} One should emphasize that there is always some freedom in the definitions of these notions. For example the large quantities $\tau, \tau' $ in the definition of a small resp. recolorable $\gb$ can be changed; they can also have an `` individual value '' $\tau_{\gb}$ ($\gg 1$) for any particular system $\gb$ etc. These ambiguitites are more important than the arbitrariness of the choice of $y$ in (3.49) but they still have no ``physical'' meaning as they will {\it not} affect the physically meaningful notion of a stable phase used in Main Theorem. \endremark \proclaim{Proposition 3} Any configuration $(x, \Cal D)$ ($\Cal D$ finite) of a mixed model containing some strictly interior small contour system contains at least one extremally small contour system. \endproclaim \demo{Proof}(This statement will be used in the proof of Main Theorem in a situation, when {\it several\/}, slightly different, mixed models appear at different levels.) Take a smallest (in $\prec$) strictly interior small contour system $\gb$ in $\Cal D$. If the collection of extremally small (equivalently, by induction, small) contour systems which are proper subsystems of $\gb$ would be empty then $\gb$ itself would be extremally small ! \enddemo As we yet announced, the importance of the notion of a small resp. extremally small contour system stems from the fact that extremally small contour systems provide practically the {\it only relevant example\/} of recolorable contour systems. Namely, the following result \footnote{One can construct, of course, examples of recolorable but not extremally small systems. However, they correspond to some marginal cases only, which are just not covered by the particular choice of the constants $\tau,\tau',\tau(\gb)$ but they are usually covered by another, more suitable choice of these constants in the definition above.} gives such a statement. It is a crucial step (together with Theorem 5 above) in the proof of forthcoming Main Theorem. \proclaim{Theorem 6} If $\hat \tau = \tau'/ 12\nu $ then any $ \tau'$--extremally small contour system is $\hat \tau$--recolorable. \endproclaim Theorem 6 will imply (we mean the values from (2.14) and (3.34)) that after the completion of the recoloring procedures of Theorem 5 (applied first to the original \ps \ model, then successively to the partially expanded models emerging from previous recoloring steps etc.), {\it no\/} small contour systems $\gb\sqsubset \Lambda$ will be left in the final mixed model in $\Lambda$. This leads directly to Main Theorem, see section 8. \endremark \vskip1mm \head {6. The proof of Theorem 6 } \endhead A contour system is $\gb'$ is called {\it tight\/} if it has {\it no\/} contour subsystem $\gb$ satisfying \footnote{Tightness of $\gb$ is the {\it same\/} notion as the tightness of its support $\supp \gb$ defined in section II.} $$ \supp \gb \sqcap (\gb' \setminus \gb) = \vv \cap \supp (\gb' \setminus \gb) = \emptyset \ \ ( \Longleftrightarrow \ \ \vv \sqsubset (\supp (\gb' \setminus \gb)^c ) .\tag 3.52 $$ If the above relation {\it is\/} satisfied we say that $\gb$ is an {\it isolated\/} contour subsystem of $\gb'$. Recall that we take the $l_{\infty}$ norm everyhere; the choice of $C=1$ instead of some other constant $C$ in $C \diam \supp \gb$ on the right hand side of (2.5) is somehow arbitrary. We will show that the proof of Theorem 6 can be reduced to the case of extremally small {\it tight\/} systems $\gb$. The smallness of $\gb$ will imply that the quantity $A(\gb)$ in $F(\gb) = E(\gb) -A(\gb)$ will become ``safely small'' with respect to $E(\gb)$. (The quantity $A^* (\gb)$ will be even {\it nonpositive\/} in the most important case of the ``stable'' external colour of $\gb$ !) The tightness of $\gb$ will then reduce Theorem 6 to some {\it combinatorial statement\/} relating the values $|\supp \gb|$ and $\conn \gb$. See Theorem 7 below. However, the case of systems $\gb$ which are {\it not tight\/} is more characteristic and more important for the proof. The quantities $A(\gb)$ (more specifically, the quantities $A^*_{\text{max}}(\square)$) will play rather important role in the announced reduction to tight systems: \definition { The skeleton } \enddefinition Let us denote the contour systems by symbols $ \Cal D$ here, the symbol $\gb$ being now used mainly for (pre)contours or even for cubes. Consider the following auxiliary construction in any volume $\Lambda$, for any configuration $y$ on $\Lambda$ which is stratified, at least locally in $\Lambda$. We will actually use it below with the special choice $\Lambda = V_{\updownarrow}(\Cal D)\setminus \supp \Cal D $ (for a contour system $\Cal D$ ) and with $y$ being the colour induced by $\Cal D$ (it is only locally from $\es$!) on $V(\Cal D)$. \definition{Construction of the skeleton} Given a stratified configuration $y$ and a cube $\square$ say that this cube $\square$ is {\it minimal nonsmall\/} if $A^*(\square , y) > \tau' ( \diam \square)$ holds for {\it some\/} extension \footnote{ Here it would be perhaps more convenient to have ``any'' instead of ``some''. Fortunately, this subtlety could be important only for cubes ``very closely sticked to $\Cal D$'' and an inspection shows that a slight increase of $E(\gb)$ in (2.14) (and therefore the increase of $\tau' = \tau -\varepsilon'$) influences the lower bound for $F(\Cal D)$ more than all these corrections.} of $y$, see (3.49) and no smaller cube which is at the same time a subcube of $\square$ satisfies such a condition. (Sometimes, for ``stable'' $y$ -- defined below in Main Theorem such a cube $\square$ will {\it not\/} exist; however this is not the case of a typical nontrivial situation below.) Given a volume $\Lambda$, let us now find some smallest possible (in $\prec$) minimal nonsmall cube $\square \sqsubset \Lambda$ (if there is some). Take all the adjacent (having distance $1$ to $\square$) cubes $\square'\ssubset \Lambda$ which are horizontal shifts of $\square$, then take all the adjacent, horizontally shifted cubes to the cubes just constructed etc. Thus we obtain some ``layer'' (only partially filled in $\Lambda$; cubes which would be too close to $\Lambda^c$ are excluded !) of cubes inside $\Lambda$. Construct also other possible partial layers {\it not intersecting\/} those constructed before, according to the rule that a layer with a {\it smallest possible\/} diameter of its ``paving blocks'' is constructed in each step. The exact meaning of the statement that the layers would not mutually ``touch'' will be that the vertical distance between any two adjacent layers is bigger than the {\it logarithm\/} of the thickness of both layers. There is of course some arbitrariness in this requirement; its reasonability will be seen below. \footnote{ The reason is to keep $\Am$ roughly additive as a function defined on the union of cubes of the skeleton. We will see below that the logarithmic distance will assure this -- because of the exponential decay of the terms $k_T$ in the sums (3.43) used in the definition of the quantities $f_t$. See Lemma below (3.54).} The collection of all minimal cubes of $\Lambda$ thus constructed will be called the {\it skeleton\/ } of $\Lambda$. We emphasise that the construction is applied in any (generally nonstandard) volume $\Lambda$ with a given {\it locally stratified\/} configuration. In particular, a {\it skeleton\/} of the ``interior''$\Lambda =\vv \setminus \supp \gb$ of any {\it extremally small system\/} $\gb$ is constructed in this way. \enddefinition \remark{Note} The fact that skeleton has a ``smallest possible grain'' is slightly superfluous here but it is useful not only below but also in other, more detailed estimates used in the study of the completeness of the phase picture constructed by Main Theorem. Notice also that in the situation where $\Lambda = \Lambda' \setminus \supp \Cal D$, $\Cal D$ extremally small, the smallest possible grain of the skeleton of $\Cal D$ guarantees that the size of any strictly interior (and therefore nonsmall) $\gb \subset \Cal D$ is {\it at least as big\/} as the distance from the nearest cube (which has a {\it smaller\/} diameter than $\gb$!) of the skeleton. \footnote{ Take the covering rectangle $\square(\gb)$, denote by $d$ its diameter. It follows from the construction of the skeleton that a {\it cube\/} of a diameter $d' \leq d$ must be already there, having a distance at most $ d'$ from $\gb$. Otherwise we could add to the skeleton, at a suitable moment of its construction, a cube of diameter $d$, touching the covering rectangle of $\gb$.} This observation is important for establishing of the {\it tightness\/} of the union of $\Cal D$ with its skeleton. \endremark We conclude: for any extremally small $\Cal D$ in a volume $\Lambda$ with a stratified boundary condition $y$ given on the boundary of the complement of $\Lambda$, we constructed the ``skeleton'' of the volume $ V_{\updownarrow}(\Cal D) \setminus \supp \Cal D $ which is composed of nonsmall {\it cubes\/}. These cubes are ``densely''packed as formulated above and the system $\Cal D$ enriched by the cubes of the skeleton is tight. \definition{Rearrangement of the (free) energy inside of the skeleton of $\Cal D$} \enddefinition Let us define the following ``rearrangement'' of the energy inside of a given extremally small contour system $\Cal D$. Imagine that the cubes of the skeleton will be treated just as some new ``contours''. The idea is to show that such an ``enrichened'' system $\Cal D^*$ of ``contours'' is {\it tight\/} in the sense formulated above (and in (2.5)) and the value of its contour functional $F(\Cal D^*)$ (see (3.53) below) is {\it smaller\/} than that of $F(\Cal D)$; however $\conn \Cal D^*$ is apparently bigger than $\conn \Cal D$. Therefore, by checking the recolorability of the enriched system we will also prove the recolorability of the {\it original\/} system. This will give the desired generalization of the argument already given in (3.51) (given there for the case when $\Cal D$, denoted as $\gb$, is a single contour. Let us show this in more detail: Denote, as announced, by $\Cal D^*$ the collection of all contours of $\Cal D$ and also of {\it all the cubes\/} of the skeleton of $\Cal D$. Let us make an agreement that for any cube $\square$, denoted here also as $\gb$, we put (just to unify the notations in the formula (3.54) below!) $|\supp \gb| = \diam \square$, $A^*(\gb)=A^*(\square,x)$ and correspondingly $$ F(\Cal D^*) = F(\Cal D) + \sum_{\square \in \Cal D} F(\square) \ \ \text{where we define}\ \ F(\square) = \tau' \diam \square - A^*(\square,x) .\tag 3.53$$ Here, $x$ denotes the configuration induced by $\Cal D$ on $\square$, (extended somehow to the whole $\zv$). With this notation, using the Peierls condition (2.14) and the definition (3.22) of $F(\Cal D)$ we can prove (3.34) by showing the following inequalities: Recall that $$A^*(\square,x) \geq \tau' \diam \square \ \ \ \text{i.e.}\ \ F(\gb) \leq 0 \ \ \text{for} \ \gb =\square $$ (this is just the negation of (3.49)) holds for any cube of the skeleton. Now we have the following relation between the contour functionals of the original extremally small system $\Cal D$ and the ``enriched''(by cubes of the skeleton) system $\Cal D^*$: (The relation below should be then written in the exponential form, to obtain the bound (3.34*); see also (3.59 below!)) \proclaim{Proposition 1} We have the relation, using the convention (3.53) and taking $\tau'$ as above $$ \tau |\supp \Cal D| - A(\Cal D) \geq \tau' |\supp \Cal D| - A^*(\Cal D) \geq \tau' |\supp \Cal D^*| - A^*(\Cal D^*) . \tag 3.54 $$ \endproclaim To prove this, notice that the first inequality just informs about the approximation of $A(\Cal D)$ by a slightly different quantity $\Af(\Cal D)$ (Proposition 1 and 2 last section). It is the {\it second\/} inequality which makes the core of this Proposition; it will be shown now to be a consequence of the {\it very definition of the skeleton\/}. Write the negation to (3.49) as $$ \tau' |\gb| - A^*(\gb) \leq 0 $$ for any new ``contour'' $\gb =\square$ of the skeleton of $\Cal D$. The idea \footnote{What follows will be just a suitable play with the quantities $\Am|\square|$ and $\tau'|\diam \square|$ (where the first quantity is replaced by the second one for any cube of the skeleton). We can interpret this replacement as an ``installing of an artificial contour $\square$ ''.} of the proof of (3.54) is that the terms $\tau'|\supp \Cal D| -A^*(\Cal D)$ are essentially {\it additive\/} as functions of the components $\gb$ of $\Cal D$. The additivity of the energies $\tau |\supp \Cal D| = \sum \tau |\supp \gb|$ (where the sum is over all contours of $\Cal D$) is of course trivial. Concerning the approximate additivity of the function $A^*$ we have the following auxiliary result. \footnote{The quantity $A(\gb)$ from section 3 is of course {\it exactly\/} additive for disjoint volumes $\vv$ but it would have some other, more severe disadvantages (than $A^*$) when the ``surface tension'' along the vertical sides of the cubes will be discussed.} \proclaim{Lemma (Approximate additivity of $A^*$)} Let $\Cal D^* = \Cal C \cup \Cal S$ be a compatible collection of contours $\Cal C =\{\gb_i\}$ and mutually disjoint cubes $\Cal S =\{\square_j\}$ such that the cubes from $\Cal S$ are not intersecting the contours of the system $\Cal C$. Let the vertical distance between any two cubes from $\Cal S$ whose projections to $\zet_{\nu -1}$ intersect be greater or equal than the logarithm of the diameter of the smaller cube. Then (we take all the quantities $\Af$ with respect to the corresponding external colour induced by $\Cal C$) $$ \Af(\Cal D^*) = \Af(\Cal C) + \sum_{\square \in \Cal S} \Af(\square ) + D(\Cal D^*) \tag 3.55 $$ where the correction term $D$ is small: It satisfies the bound, with a large $\tilde \tau(\tau)$ $$ |D(\Cal D^*)| \leq \sum_{\square \in \Cal S}( \diam \square)^{-\tilde \tau} .$$ \endproclaim \demo{Proof} Consider two such collections $\Cal D^*$ which differ just by {\it one cube\/} $\square$ \ i.e. let we also have a bigger collection \ $\Cal D^{**} = \Cal D^* \& \square $. Assume that the cube $\square$ is not greater than any cube of $\Cal D^*$. It is now sufficient to prove the bound $$ |\Af(\Cal D^{**}) -\Af(\Cal D^*) -\Af(\square)| \leq (\diam \square)^{- \tilde \tau}. \tag 3.55' $$ Notice that the quantity $f_t$ in (3.43) can be written also as the sum over intervals $I$ $$ f_t(y) = e_t(y) - \sum_{I \subset \zet\ : \ t \in I} k_I $$ where $k_I$ is the sum of all the contributions to (3.43) having a fixed projection $I$ of $T$ to the last coordinate axis. Of course, we have a bound, for suitable large $\hat \tau$ $$|k_I| \leq \exp(-\hat \tau |I|).$$ Imagine that in the expression on the left hand side of (3.55') we {\it ignore\/} (when substituting these quantities, for any vertical section of $\square$ and any $t$, into (3.47)) {\it all\/} $I$ intersecting {\it both\/} $\square$ and some {\it other\/} cube resp. contour of $ D^*$. Then the relation (an analogy to (3.55')) $$ A^*_{\text{ignore}}(\Cal D^{**}) - A^*_{\text{ignore}}(\Cal D^*) - A^*_{\text{ignore}}(\square) = 0 $$ (we use the subscript``ignore'' instead of ``full'' to denote the quantities just mentioned) is {\it exact\/}, as simple inspection shows. The correction due to the quantities $k_I$ just ignored is then obviously of the order $\exp(-\hat \tau d) $ where $d$ is the distance of $\square$ and the cubes from $\Cal D^*$. This proves (3.55'), and therefore also (3.55). \enddemo Let us continue now the investigation of the right hand side of (3.54): By (3.50), the right hand side of the relation (3.54) is {\it greater\/} than (we use here the very smallness \footnote{ Notice that our use of the {\it squares\/} in the definition of smallness is rather important technically. Namely, our method of the proof here relies quite heavily on the fact that the enriched (by cubes of the skeleton) system $\Cal D^*$ of any small $\Cal D$ is again a small system! Our very definition of smallness using cubes gives automatically such a property; otherwise much more cumbersome (though possibly ``physically more intuitive'') notion of a smallness would be needed.} of the cube $\square(\Cal D^*)$ !) $$ \tau'|\supp \Cal D^*| -\Af(\square(\Cal D^*)) \geq \tau' (|\supp \Cal D^*| - \diam \square(\Cal D^*)) \tag 3.56 $$ which is surely {\it greater\/} (notice that $ \diam \square(\Cal D^*)= \diam \square(\Cal D)$!) than, say (see the footnote below (3.59)), $\tau /2 \ |\supp \Cal D^*|$. Thus it suffices to show now that the quantity $ \tau /2 \ |\supp \Cal D^*|$ (and, therefore, also the right hand side of (3.56)) is greater than, say, \ $ \tau /12\nu \ \conn \Cal D$. We will prove this by proving the following result (Theorem 7). Generalize again the notion of a tight system $\Cal D$ to any subset of $\zv$ (and compare with (3.52)): \definition{Tight sets} \enddefinition We will relax here slightly, for the purposes of the forthcoming purely topological section, our definition of a tightness i.e. the relation $\sqcap$ on which the notion of tightness of contour systems was based: Now we will say that $S \subset T$ is isolated in $T$ (where $ T \subset \zv $) if we simply have $$ \dist(S, T \setminus S) \geq \diam S. \tag 3.57$$ Say that $T$ is tight if there are no isolated subsets of $T$. \remark{Notes} 1. The very concept of $ \vv$ above (starting from (2.16)) was motivated by our introduction of the notion of a {\it strict\/} interiority and diluteness. Topologically, there is not much difference between the notions of tightness of $\Cal D$ defined in terms of tightness of {\it contours\/} of $\Cal D$ (as everywhere before, which was more convenient in Theorems 3, 4, 5) and tightness of $\supp \Cal D$ defined simply in terms of (3.57) (which is slightly weaker requirement, as simple inspection shows). \newline 2. Notice that the system $\supp \gb^*$ is already tight (even in the sense of $\sqcap$) because the definition of a skeleton of $\gb$ gives no room for isolated (in the sense of empty $\sqcap$) subsystems for the enriched set $T = \supp \gb^* $ ! \endremark \proclaim {Theorem 7} If $T$ is tight then $$ \conn T \leq 6\nu |T| . \tag 3.58$$ \endproclaim Apparently, the proof of this relation will finish also the proof of Theorem 6. Namely, we can conclude the arguments of (3.54) and (3.56) as follows: $$ \tau |\supp \Cal D| - A(\Cal D) \geq \tau' |\supp \Cal D^*| - \Af(\Cal D^*) \geq \tau'|\supp \Cal D^*| - \tau' \diam \square(\Cal D^*) \tag 3.59 $$ i.e. we finally get, using a bound $|\supp \Cal D| > 2 \diam \square(\Cal D^*)$ \footnote{We omit the proof of this geometrical statement (which is trivial for ``nonexotic'' systems $\Cal D$).} and (3.58), the relation $$ \tau |\supp \Cal D| - A(\Cal D) \geq {\tau \over 2} |\supp \Cal D^*| \geq {\tau \over 12\nu} \conn \Cal D \tag 3.59 $$ which gives the {\it recolorability\/} (3.34) of the system $\Cal D$. Let us now start the {\it proof of Theorem 7\/} by restricting it first the the case of sufficiently big cardinalities only. (This assumption is used in Lemma 1, section 7 below.) First consider the case of sets of cardinality, say, at most $1024 = 2^{10}$. We will check the bound (3.58) for any tight set $T$ whose cardinality is $d \leq 1024$. More generally we will prove the following weaker analogy of (3.58), which is valid for all \footnote{We concentrate on $\nu =3$ and we do not try to optimize all the estimates given here and below.} $d$: $$ \conn \Cal D \leq 3 d \ \log_4 d \ ,\ \ \ d = \card \supp \Cal D \tag 3.60$$ We will prove this by the induction, over the number of connected components $\{T_i\}$ ; $ i =1,\dots,n$ \ of the set $T$. The case $n=1$ is trivial. Take the smallest component, say $T_1$, of $T $ . Notice that no more than $3 d_1$ unit steps are necessary to connect $T_1$ with the rest of $T$ by a connected path (having unit steps). (We use here the fact that $T_1$ is {\it not\/} isolated in $T$, compare (3.52), and assume first for the simplicity that $T\setminus T_1$ is tight, see below.) Thus we have to check the following inequality, where $d_2 $ denotes the cardinality of $T \setminus T_1$: $$ d_1 \ \log_4 d_1 + d_2 \ \log_4 d_2 +d_1 \leq (d_1 + d_2) \ \log_4 (d_1 + d_2). $$ This inequality is proven by elementary analytical calculation, writing $d_1 = \alpha d_2$ where $\alpha \in (0,1)$. If $T\setminus T_1$ consists of {\it several\/} tight components $T_2,\dots,T_m$ then we need to connect $T_1$ to each of them i.e. instead of the above inequality we need a generalization $$ (m-1) d_1 \ + \sum_{k \geq 1}d_k \ \log_4 d_k \leq (\sum_{k \geq 1} d_k) \ \log_4 (\sum_{k \leq 1} d_k) $$ which is proven easily by induction over $m$ (recall that the cardinality $d_1 =\card T_1$ is assumed to be smallest from the numbers $d_k =\card T_k$). Thus it really suffices to prove the desired inequality $ \conn T \leq 6 \nu|T| $ (see (3.58); we need it for $T = \gb^*$) at the assumption that $\supp \gb^*$ has a cardinality at least $1024$: \definition {Proof of (3.58) for large $T$. Commensurately connected collections of cubes} \enddefinition It will be useful to {\it reformulate\/} the notion of a commensurately connected graph to another language based on the employment of {\it cubes\/} from $\zv$ instead of bonds from $2^k \zv $: \definition{Definition} Say that the two cubes $\square , \square' \subset \zv $ are commensurate if $$ \square \cap \square' \ne \emptyset \ \ \ \text{and} \ \ \ |\log_2 \diam \square - \log_2 \diam \square' | \ \leq \ 2 . \tag 3.61 $$ \enddefinition (Notice that the constant $2^1$ in (3.2) was replaced by $2^2$ in (3.61). This is for purely technical reasons and will be convenient below.) \proclaim {Proposition 2} If \ $G$ is a commensurately connected {\it graph\/} then the collection $\{\square(b), b \in G \}$ of covering {\it cubes\/} of bonds of $G$ is commensurately connected in the sense above. \endproclaim This is immediate, by comparing (3.2) and (3.61). The opposite relation (that any commensurately connected collection of cubes can be ``approximated'' by a commensurately connected graph) can be also established: Introduce first another auxiliary geometrical notion. \definition{Definition} If $\square$ is a cube with a diameter $2^k \leq \diam \square < 2^{k+1}$ then the lexicographically first point of $\square \cap 2^k \zv$ will be called the anchor of $\square$, denoted by $ a(\square)$. \enddefinition \proclaim {Proposition 3} Let $\Cal S = \{\square_i\}$ be a commensurately connected collection of cubes from $\zv$. Then there is a commensurately connected tree $\Cal T$ such that all the anchors $a(\square_i)$ are among the (possibly multiple) vertices of $\Cal T$ and $$ |\Cal T| \leq 3 \nu |\Cal S| . \tag 3.62 $$ \endproclaim \demo{Proof} We may assume that $\Cal S$ is already a tree. Take any commensurate bond $\{\square, \square'\} \ \in \Cal S$. Write $[\log_2\diam\square] = k, [\log_2\diam \square'] = k' $; we may assume that $k' \in \{k, k+1, k+2 \}$. A straightforward inspection shows that it suffices to consider the case $\nu = 1, k = 1$, and $ a(\square') = 0 $. Notice that then the following path from $a = a(\square) $ to $ a' = a(\square')$ can always be constructed : \ \ $ a' = a + v_1 + v_2 + v_3 $ \ \ where the vectors $v_i$ having the lengths $2^{l_i} ;\ l_i \in \en$ satisfy the following requirements: $$ k \leq l_1 \leq k+1, \ l_1 -1 \leq l_2 \leq l_1 + 1 ,\ l_2 -1 \leq l_3 \leq l_2 +1 , \ k' \leq l_3 \leq k' + 1 .$$ It is clear that the tree defined by all the bonds \ $\{a,a+v_1\},\{a+v_1,a+v_1+v_2\},\{a+v_1 +v_2, a'\}$ (where $a, a'$ vary over all commensurate pairs $\square,\square'$ and the triple above is repeated in the direction of any coordinate axis) is commensurately connected. \enddemo \proclaim{Proposition 4} Let $S \subset \zv $. Let $\conn_{\square}$ denote the cardinality of a smallest commensurately connected collection of cubes satisfying the following requirement: if all points of $S$ are added (we identify the points of $\zv$ as cubes of diameter $1$) then the whole collection is commensurately connected. Then $$ \conn_{\square} S \geq { 1 \over 3\nu} (\conn S). \tag 3.63 $$ \endproclaim We will now prove Theorem 7 by showing that the inequality $$ \conn_{\square} S \leq |S| \tag 3.64 $$ holds for any tight set $S$ whose components have a cardinality at least $1024$. \vskip1mm \head 7. The Proof of (3.64) \endhead Let us make an agreement that an explicit choice of the appropriate constants here will be given below only for the case of the dimension $\nu = 3$. (Apparently, for $\nu > 3$ the final constant in Theorem 7 is even better -- but we do not care here.) Define a suitable collection of cubes having a size $2^k $ where $k =1, 2, 3, \dots $ such that any cube in $ \zv $ can be ``packed'', with a reasonable ``accuracy'', by some cube of the collection: \definition{ Second covering cube } Denote by $ \Cal K_k$ the collection of all cubes in $\zv$ which are shifts, by suitable values from the lattice $2^{k-1}\zv$, of the unit cube $[0,2^k]^{\nu} $ in the lattice $2^k \zv $. Write $ \Cal K = \cup_k \Cal K_k$ where $k= 1,2,\dots $. We have the natural ordering $\prec$ on $\Cal K$ extending the ordering by size resp. the lexicographic ordering of shifts of one particular cube; this ordering can be extended to suitable total ordering of {\it all\/} cubes in $\zv$ which is in accordance with the inclusion relation as well as with the lexicographic order of mutually shifted cubes, and we denote by $\widehat{\square}$ the (lexicographically first) cube from $\Cal K_k$, $k$ smallest possible, containing $\square$. This will be called the {\it second covering cube\/} of $\square$ resp. of a set $S$ such that $\square = \square(S)$. \enddefinition Notice the following fact : if $\square$ is the covering cube of $S$ then the second covering cube of $S$ contains $\square$ and has a diameter at most four times bigger than $\square$. \definition {Black and grey cubes of a set $ S \subset \zv$ } Say that a cube $ \square$ is a black cube of $ S \subset \zv$ if $ \square \cap S $ contains at least $4 (\diam \square)^{1 \over 2}$ points, resp. at least $\diam \square$ points if its diameter is smaller than $16$. Any cube {\it from $\Cal K$\/} which is the second covering cube of some black cube of $S$ will be called the grey cube of $S$. (It obviously contains at least $2 (d')^{1\over 2}$ points of $S$ where $d'\leq 4d $ denotes the diameter of the corresponding grey resp. black cube.) We will show the following statements : \enddefinition \proclaim{Lemma 1} If \ $T$ is tight and its connected components have a diameter at least $16$ then the collection of its black cubes is commensurately connected. \endproclaim \proclaim{Lemma 2} The number of grey cubes of a size at least $1024$ of any set $S \subset \zv$ is no greater than $1/2 \ |S|$. \endproclaim \proclaim{Lemma 3} If $\Cal S = \{\square_i\}$ is a commensurately collected collection (of black cubes of some set $S$) then the collection $\widehat{\Cal S} = \{ \widehat{\square_i } \}$ (of grey cubes of $ S$) is contained in some commensurately connected collection $\Cal S'$ such that $ |\Cal S'| < 2 |\widehat{\Cal S} |$. \endproclaim \demo{Proof of Lemma 1} We will proceed by the induction over the number of points in $T$. Say that $S$ is a {\it nice\/} subset of $T$ if it has the following property : for any $t \in S$ there is a commensurately connected collection of black cubes $\{\square_i\}$ of $S$ which is concentric (i.e. $t \in \square_i$ for each $i$ ), starts in the covering cube of $S$ and ends in $t$. Take some maximal nice subset $S$ of $T$. We will show that $S =T$\ if \ $T$\ is tight. Really, if $ N =T \setminus S $ is nonempty then either there is some isolated subset $M$ of $N$ or $N$ is tight. In the former case take $M$ as the smallest possible isolated (and therefore tight) subset of $N$. Then $M$ (or $N$ itself, in the latter case) is also nice by the induction assumption. Take the covering cube $\square(M)$ of $M$. We claim that there is some black cube $\square'$ of $S$ such that $$ \dist(\square(M),\square') < \diam \square(M) \ \ \text{and} \ \ | \log_2 \diam \square(M) - \log_2 \diam \square'| \leq 1 . \tag 3.65$$ This follows from the fact that $M$ can{\it not\/} be isolated in $S\cup M$ (otherwise $M$ would be isolated also in $T$ ). Therefore, there is some $t \in S$ whose distance from $\square(M)$ is no greater than $\diam \square(M)$ and we take an appropriately large black cube $\square' \ni t$. (Its existence is guaranteed by the ``nicety'' of $S$.) Now, if $\square^*$ is any black supercube of $\square'$ (black in $S$ ) then the supercube of $\square(M) \cup \square^*$ -- denoted as $\square^{**}$--will be shown to be again a black cube (of the whole set $T$) and this would mean that $S \cup N$ would be nice, as simple inspection shows.(Check that there is now a commensurate path from $t$ to $\square(M)$ and any commensurate chain of cubes going ``up'' from $\square(M)$ through cubes of the type $\square^*$ can be modified by going through corresponding cubes $\square^{**}$. Thus $ S =T$. The modified chain is clearly also a commensurate one, containing $\square(M)$.) The observation that $\square^{**}$ is black in $T$ follows from the following more general \proclaim {Lemma 4} If \ $\square', \square'' $ ($\square'' = \square(M) $ in the above application) are two cubes which are black cubes of some sets $ T' ,\ T'' \ ; \ T' \cap \ T'' = \emptyset$ and such that $\dist(\square',\square'') \leq \diam \square'$ and $ |\log_2 \diam \square'' - \log_2 \diam \square'| \leq 1$ then the covering cube $\square (\square' \cup \square'') $ is the black cube of the set $ T = T' \cup T'' $. \endproclaim The validity of Lemma 4 is easily seen (it suffices to consider the case of the dimension $\nu = 1$!) from the inequality ($d$ denotes the distance between two cubes $\square'$, $\square''$; the diameter of $\square'$ is assumed to be $1$ while the diameter of $\square''$ is denoted by $x$) $$ d \leq 1 \ \& \ 1/2 \leq x \leq 2 \Rightarrow (1 + x + d)^{1 \over 2} \leq 1 + x^{1 \over 2}. \tag 3.66$$ \enddemo \demo {Proof of Lemma 2} We will give the proof only for the case $\nu = 3 $. For the purposes of this proof modify the cubes $a + [0,2^k]^{\nu} $ from $\Cal K_k $ to the following form: $ a + [0, 2^k -1 ]^{\nu}$. Then the system $\Cal K_k$ can be decomposed into $8$ pavings of $\zv$ by disjoint sets. Take the sum $${8 \over 2}\sum_{10}^{\infty} {1 \over 2^{k \over 2}} < 7/16 \tag 3.70 $$ and imagine that any point of $t \in S $ transfers the ${1 \over 2} 2^{-k \over 2}$ -- th portion of its ``unit mass'' (3.70) to any cube of $\Cal K_k$ containing $t$. By the definition of a grey cube, the total mass thus transferred to any grey cube of $S$ is at least $1$ and, therefore, the cardinality of the set of all grey cubes of $S$ (of a diameter bigger than $1024$) is smaller than $7/16\ |S| \leq 1/2 \ |S|$. \enddemo \demo{Proof of Lemma 3} This easily follows from the following observation : if $\square, \square'$ are commensurate cubes then either their second covering cubes $\widehat{\square},\widehat{\square'}$ are also commensurate or one of these latter two cubes can be replaced by an auxiliary, twice bigger supercube from $\Cal K$ such that the first statement is true. \enddemo This concludes the proof of Theorem 7 if we moreover notice, that to connect all the components of $S$ to their black supercubes we can construct commensurate paths with less than, say, $1/2 \ |S|$ steps. Thus, also Theorem 6 is proven. \remark{Note (on the applicability of the notion of a skeleton to a situation of \cite{Z})} Our new Theorem 6 is a stronger and better replacement for the ``Main Lemma'' of [Z]. Really, one can use its (simpler) analogy also in the translation invariant situation of [Z]. Then it can have (e.g.) the following form: If $\gb_i$ are mutually external ``large contour systems'' satisfying (in the notations of \cite{Z}) the relation $a_q |V(\gb_i^q)| > \tau |\supp \gb_i^q|$ then $$ a_q |\ext| > \tau \Conn(\Lambda,\{\gb_i^q\}) \ \ \ \text{where}\ \ \ \ \ext = \Lambda \setminus \cup_i V(\gb_i),\tag 3.71 $$ the integer $\Conn(\Lambda,\{\gb_i^q\}$ on the right hand side denoting the cardinality of a smallest possible set whose union with $\Lambda^c$ and all $\supp \gb_i$ is {\it connected\/}. With this lemma, one can rewrite the usual \ps \ theory in a way analogous to that used here without an explicit construction of the contour models. See the lecture notes \cite{ZRO} for more details. \endremark \vskip1mm \head 8. Total Expansion of a General \ps \ Model. The Metastable Model. The Main Theorem. \endhead Up to now we worked with a general mixed model, though having in mind that the constructions developed above should be used for a {\it study of successive expansions of the original\/} \ps \ model. To summarize our previous investigations: Theorem 6 enables to {\it repeat\/} the algorithm of {\it expansion\/} formulated by Theorem 5, again and again. \footnote{ One can start more generally with a suitable mixed model: we will see that this remark is important for some classes of {\it infinite range\/} models.} \definition{ Removable contour systems of a general mixed model} A recolorable ($\hat \tau$ -- recolorable!) contour system $\gb$ which is a strictly interior contour subsystem of some configuration $\Cal D$ will be called {\it removable\/} if it is moreover smallest possible one in $\prec \prec$. (We announced this notion already in Section 5; recall that a basic example of a $\hat \tau $-- recolorable system is a $\tau'$ --{\it extremally small\/} one.) In a fixed volume $\Lambda$ we always accompany the above condition by the requirement that $\gb$ would be also isolated (see (3.52)) from $\Lambda^c $. This is in accordance with the requirement $\gb \sqsubset \Lambda$ used in our definition of $Z_{\updownarrow}(\Lambda)$. \enddefinition \definition{Iterative use of Theorem 5. The totally expanded \ps \ model} It is now important to realize that the process of recoloring (of the collection of shifts of a removable $\gb$) formulated by Theorem 5 can be applied again to all the newly emerged (after all their strictly interior recolorable subsystems being ``swept'' by previous applications of Theorem 5), removable contour systems. \footnote{Such new removable systems appear as recolorable ``rudiments'' of contour systems $\gb$ in the situation when all the recolorable subsystems $\gb' \subsetneqq \gb$ were already expanded by the previous applications of Theorem 5. Notice that in the course of this successive construction, due to the appearance of the new quantities $k_T(y)$, occasionaly some previously recolorable (but too big to be removed at that moment) objects may turn out later (at the moment they {\it could\/} be removed) to be nonrecolorable. Also, some previously nonrecolorable objects $\gb$ can be recolorable later from the point of view of some later expanded model-- but we do {\it not\/} remove them yet, to avoid confusion in the construction of clusters $T$ (using relation $\prec$). These marginal phenomena do not affect the sense of our contruction. Let us stress once again that only the ``smallest possible recolorable contour systems'' are recolored at any stage of the expansion.} This iterative process can be therefore repeated up to the very moment when there are {\it no\/} removable, equivalently {\it no\/} ($\hat \tau$--)recolorable, in particular {\it no ($\tau'$--)extremally small\/} (by Theorem 6) strictly interior contour subsystems in the ``final'' mixed model, where ``where nothing recolorable remains''. This final mixed model will be called the {\it totally expanded \ps \ model\/} corresponding to the abstract Pirogov -- Sinai model given by Hamiltonian (2.12) and satisfying (2.14). \enddefinition \definition{$ G$ -- expanded \ps \ model} This is the result of successive applications of Theorem 5 (see above) performed up to the moment when no $G$ --removable \footnote{Some (marginal cases of) newly $G$--recolorable contour systems $\gb$ which were $G'$--nonrecolorable, $G' \subset G$ in previous expansion steps may appear, as we noted above. We do {\it not\/} recolor them yet.} $\gb$, $\supp \gb \prec G$ remains. (This operation is done simultaneously in all finite $\Lambda$, for any boundary condition $y \in \es$.) The following result is then a direct consequence of the very construction of the totally expanded model. It is of a crucial importance in the following. \enddefinition \proclaim{Corollary (of the definition of a totally expanded model)} In the totally expanded model, only those contour subsystems remain which are not recolorable (therefore: not small) and contain no recolorable contour subsystem. \endproclaim \definition{Residual systems} A contour system $\gb$ which exist as a configuration of the totally expanded model (equivalently, $\supp \gb$ -- expanded model) \footnote{ Residuality is decided on the level of $\supp \gb$ -- expanded model, and later expansion do not matter!} will be called a {\it residual\/} one. If \ $\tilde \Cal D$ is a contour system in $\Lambda$, under some (unstable) boundary condition $z \in \es$ remaining from $\Cal D$ after the total expansion of the model we will say that $\tilde \Cal D$ is a {\it residuum of \/} $\Cal D$. \enddefinition \remark{Notes} 1. An equivalent characterization of residuality in terms of the related notion of a metastability will be given below. Notice that the removability in a volume $\Lambda$ of $\gb$ means just the recolorability of $\gb$ \& {\it residuality\/} of all the isolated contour {\it sub\/}systems of $\gb$ \& isolation of $\gb$ from $\Lambda^c$ if we are in a volume $\Lambda$ . Speaking about recolorability, residuality etc. of some $\gb$ in Corollary above one has in mind the corresponding property of $\gb$ in the ``provisional'' mixed model, constructed up to the moment when these properties of $\gb$ are checked (just before applying Theorem 5 in the case of removability). In this provisional mixed model, there are no more any removable systems $\gb',\gb' \prec \prec \gb$. 2. In fact, (non)recolorability or (non)smallness of a contour system $\gb$ in the final, totally expanded model and in the provisional one mean almost the same. (However, only the {\it latter\/} notion is used!) More precisely we note that for cubes $\square$ of a comparable (or smaller) size as $\gb$, the quantities $A^*(\gb)$ and $A^*(\square)$ are almost the same for both the provisional (expanded ``up to the size of $\gb$'') and the final expanded model. (Notice that the quantities $A(\gb')$ are {\it exactly\/} the same, in both mixed models, for contour systems satisfying the condition $\diam \gb' \leq \diam \gb$.) More precisely, clusters $T$ having a {\it different\/} value in considered mixed models (the provisional one and the final one) have a size {\it at least\/} $\diam \gb$ and thus the difference between the corresponding values of $A(\gb)$ discussed above is of the order $\varepsilon^{\diam \gb}$; a tiny quantity compared to $\tau |\supp \gb|$! This fact, together with the fact that that $A^*$ and $A$ are almost the same in any mixed model, will be used later in the proof of Main Theorem (relation (3.81)) and also in (3.90) and below it. \endremark \definition{Metastable model} A configuration $(x_{\Cal D}^{\text{best}},\Cal D) $ which is $y$ -- diluted, \ i.e. equal to $y \in \Cal S$ outside of some set having standard finite components, will be called {\it $y$ -- th metastable\/}, shortly metastable, if there is a sequence of successive removals (expansions by Theorem 5) of removable subsystems (of the remainders of \ $\Cal D$\ left by the preceding applications of Theorem 5) leaving an {\it empty set\/} at last. \footnote{ In contrary to the usual applications of the \ps \ theory we do {\it not\/} try to express such a property in terms of ``external contours'' of the system. Such a characterization would be cumbersome.} The restriction of the original abstract Pirogov -- Sinai model (with the Hamiltonian (2.12)) to all $y$ -- th metastable configurations will be called the {\it $y$ -- th metastable model\/}. If the abstract model was constructed as a representation of an original model (2.2), then we define the $y$ -- th metastable submodel of (2.2) as the restriction of the Hamiltonian (2.2) to the configuration space $\ex^y_{\text{meta}}$ of all diluted configurations $x \in \ex$ whose representations $x \equiv (x_{\Cal D}^{\text{best}},\Cal D)$ in terms of contours are $y$--th metastable in the sense above. We can also say that $\Cal D$ is {\it residual\/} if and only if it has {\it no\/} isolated metastable subsystems. Now we are able to formulate our {\it basic result\/}: \enddefinition \definition{The free energy of the expanded model} Recall the quantities $f_t(y)$, $y\in\Cal S$ which were defined in (3.43) for {\it any\/} abstract mixed model. Consider now these quantities for the case of the totally expanded model whose construction (starting from the original \ps \ model) was just described. In such a case, the quantity $f_t(y)$ will be denoted by \footnote{To emphasise the fact that $k_T(y)$ are not simply just some additional small quantities added to $e_t$ and $E(\gb)$ but some {\it very special functions\/} of the original abstract \ps \ model.} $$ h_t(y) \equiv \ f_t(y) . \tag 3.72 $$ \enddefinition \definition{Ground states of the totally expanded model: minimalization of $\{h_t(\cdot)\}$} Say that $y \in \es$ is a ground state of the totally expanded model if the inequality ($[t]$ denotes the collection of all $t'= t+ (0,0,\dots,0,t_{\nu}); \ t_{\nu} \in \zet $) \footnote{Obviously, this sum does not depend on the choice of $t \in \zv$.} $$ \sum_{t'\in[\,t\,]}(h_{t'}(\tilde y)-h_{t'}(y) ) \geq 0 \tag 3.73 $$ holds for any $\tilde y \in \es$ which differs from $y$ on a layer of a finite width only. \enddefinition \definition{Stable elements of $\Cal S$} A stratified configuration $y\in\Cal S$ for which there is {\it no\/} residual contour system $\gb$ such that $(x_{\gb})^{\ext}= y$ will be called {\it stable\/}. In other words, $y$ is stable if the collection of configurations $ x\neq y$ having the value $y$ outside $\Lambda$ is empty in the totally expanded model, for {\it any\/} finite $\Lambda$. This notion will be used only under additional (very natural, see the discussion below (3.94)) assumption that contours of an {\it arbitrarily large diameter\/} exist in the original \ps \ model, for any external colour $y \in \es$. \enddefinition \proclaim{Main Theorem} Consider an abstract Pirogov -- Sinai model (section II, 6) defined by Hamiltonian (2.12), (2.14), with a large $\tau$. Moreover, assume that contours of all sizes exist, for any external colour $y \in \es$. Then the stable configurations $y \in \es$ are precisely those configurations from $\es$ which are the ground states of $ h_{t}(\cdot) $. \footnote{The quantities $h_t(y)$ can be interpreted as the densities, at the point $t \in \zv$, of the free energy of the corresponding $y$-- th metastable model. To understand their meaning, see e.g. (2.26).} Moreover, for any stable $y $ there exists an ``abstract Gibbs measure'' $P^y$ on the configuration space $$X^{\text{abstract}}_{\Cal I,y} \subset X^{\text{abstract}}_{\text{fin}} \tag 3.74$$ (see Section II, 7) of the given abstract \ps\ model whose almost all configurations are $y$--th metastable. The conditional finite volume probabilities $P^y_{\Lambda}$, $y$ stable, conditioning of $P^y$ being taken with respect to strictly $y$--diluted configurations in $\Lambda$, correspond to the Gibbs probabilities on the ensembles $X^{\text{abstract}}_{\Lambda,y}$. \endproclaim For the Hamiltonian (2.2), we get the following straightforward consequence, discussed in more detail in Section 9 (see the formula (3.93) and the discussion of more general boundary conditions below it) of the paper. \proclaim{Corollary} If the abstract Pirogov -- Sinai model of Main Theorem represents some ``physical'' model given by Hamiltonian (2.2), then for any stable $y $ there exists a Gibbs state of the model (2.2) on $\ex = S^{\zv}$ whose almost all configurations can be identified, in the contour representation $x_{\Lambda} = (x^{\text{best}},\gb)$, with a suitable element of $ X^{\text{abstract, meta}}_{\Cal I,y}$ of all $\Cal I$ -- finite, $y$--th metastable configurations. \footnote{One could prove also the exponential decay of correlations in any (metastable) Gibbs state thus constructed. We omit the details. See, however, Section 9 for some information explaining, or at least preparing the ground (relation (3.93)) for the proof of these facts.} \endproclaim \remark{Notes} {\bf 0. } Clearly, the families of configurations $\ex_{\text{meta}}^y$ are mutually disjoint for different $y \in \es$. We will not study here in much detail the structure of a typical configuration of the ``$q$ -- th Gibbs state''. See, however, the final section 9 for some information. (This problem deserves a detailed treatment. However, it seems reasonable to do this in connection with an investigation of other related questions, like the completeness of the phase picture constructed here. We plan to devote a separate paper to these questions.) \newline {\bf 1. } There are no other stratified Gibbs states of such an abstract model. We are not giving here the proof of such a completeness of our phase picture (characterized by the stable values of $y \in \Cal S $). It can be done similarly as in \cite{Z}. See also some comments in the section Concluding Notes below. However, we plan a more systematic treatment of this and related questions in a separate paper.\newline {\bf 2. } By the phrase that ``the $y$ -- th Gibbs state can be identified with the corresponding $y$ -- th metastable model'' we mean that almost all configurations of this Gibbs state are $y$ -- diluted and moreover the ``islands''(say the components of $\vv$ where $\gb $ is the collection of all contours of the the considered configuration $x$; $\vv$ the set of all points of $x$ which are {\it not\/} $y$ -- stratified) are typically ``small'' and ``rare'', but distributed with a uniform density throughout $\zv$. For a more complete statement, see the section 9 below. \newline {\bf 3.} Having in mind that the quantities $h_t(y)$ can be effectively computed from the expansions (3.43) (within a given precision; of course in full this is a horribly complicated sum -- but its terms are converging {\it very\/} quickly) our Main Theorem gives, in fact, a {\it constructive criterion\/} for finding the stable values $y\in\Cal S$. Practically, one may suggest an approximate finding of stable values of $y$ from some auxiliary ``$M$--expanded'' model ($M$ is some square, for example) where only those removable contours whose size does {\it not\/} exceed the size of $M$ are recolored. In fact, even for squares $M$ quite small some useful approximations can be found, often enabling yet to distinguish between the stable and nonstable $y$. This is because the series (3.43) are really very quickly converging and moreover we often have some additional symmetry in the special cases of interest; like the $+/-$ symmetry in some special cases of the Blume Capel models. For those models, even the smallest size 2$\times$2 of the square $M$ can be useful -- see \cite{BS}. Namely, considering only first few terms in (3.43) a correct conclusion about the stability of the $0$ phase resp. the $+/-$ phase can be made. \newline {\bf 4.} In fact, in ``reasonably shaped'' (see below) finite volumes $\Lambda $ there is no noticeable difference between the behaviour of the stable $y$ --th phase and another $\tilde y$ --th phase if $$ a=\sum_{t\in[\,t\,]}(h_{t}(\tilde y)-h_{t}(y) ) \tag 3.75 $$ satisfies, say, \ $\tau a^{-1} >(\diam \Lambda)^{\nu-2} $. Quite straightforward estimate of the quantities $$ A(\square) \leq a |p(\square))| \leq a (\diam \square)^{\nu- 1} \leq \tau \diam \square \tag 3.76 $$ where $p$ denotes the orthogonal projection on $\zw\subset \zv$ shows that the equations of the type (3.49) cannot be violated in such a small volume. ``Small'' can have a meaning ``having a diameter of the order $10^{27}$'' here. Namely, consider the situation (quite typical in applications) when we have some apriori given subset $\es' \subset \es $ of ``all the reasonable candidates for almost ground states'' -- such that we are sure that no stable $y$ could be found outside of $\es'$. In such a case the size of the quantity $a$, more generally the size of the difference between various $\sum_{t= (t_1,\dots,t_{\nu -1},(\cdot)} h_t(y), \ y \in \es'$ is only of the order $a_G +a_P \diam \Lambda$ where $a_G$ is the maximal deviation between the ``ground''sums (over the vertical lines $\{t= (t_1,\dots,t_{\nu -1},(\cdot))\}$) $\sum_{t= (t_1,\dots,t_{\nu -1},(\cdot))} ( e_t(y)$, $y \in \es'$ while $a_P$ is the maximum of the difference between values on the left hand side of the exponential Peierls bound (2.14) given for the same $G =\supp \gb$ but for the different external colours $y \in \es'$. If we take e.g. a perturbed Ising model with a horizontally invariant external field whose ground vertical sums do not exceed a value $\lambda$ then one can estimate that $a_P$ is of the order $\exp(-n \beta J) \lambda$\ (where $n > 4 $ or even more) and apparently one does not need to have $\lambda$ very small to reach that, say $a_P < 10^{-100}$ for {\it small\/} temperatures $\beta^{-1}$. So the question of stability in such a volume is often just a question of behaviour of the ``ground'' part of $h_t(y)$ i.e. of finding of minimum of the vertical sum of $e_t(y)$, within a given precision. \newline {\bf 5.} Though the question of the existence of {\it at least one\/} stable $y\in \Cal S$ is not the absolutely crucial one (as the preceding note shows) one should mention, nevertheless, that at least one stable $y\in\Cal S $ really {\it does exist\/}: Say that $y\in\Cal S $ is $N$-ground if for any $\tilde y\in \Cal S $ such that $\tilde y=y$ for $|t_\nu|\geq N$ we have the inequality (3.73). Now, if the configurations $y^{N}\in\Cal S $ are $N$-ground then a suitable subsequence of $\{y^{N}\}$ must converge to some ground (``stable'') value $y\in\Cal S $. (We use the compactness of the space $\Cal S $ in this argument as well as the quick convergence of the cluster series for the quantities $h_t(y)$). \newline {\bf 6.} For some models, like the Ising model with stratified random external field, the collection of all (almost) stable $y\in\Cal S$ can be {\it very rich\/} and the phase diagram -- as the function of all (vertically dependent) values of the field -- extremely complicated. We plan to study this particular case in some later paper. \newline {\bf 7.} The latter example case shows that it is not very reasonable to try to formulate results about the shape of the {\it phase diagram\/} in full generality here -- because possibly infinite parameters are present in the Hamiltonians of the stratified type. However, if we call by a phase diagram of the model the very {\it mapping\/} $$ \{ \ \ y \in \Cal S \ \longmapsto \{ h_t(y) \} \ \ \} \tag 3.79 $$ then the information about the actual phase diagram, its dependence on the parameters in the Hamiltonian etc. can be deduced from (3.79) ; just by using suitable variants of the implicit function theorem (possibly with infinitely many variables). However, this paper is not on analysis of manifolds and so we omit these questions completely. It is worth noticing here that, in order to get a best possible smoothness of the mapping (3.79) (and of the mappings derived from it by implicit function theorems), it may be reasonable to {\it modify\/} suitably the definitions of extremally small contours etc. -- to obtain the best available differentiability (even local analyticity) properties of this mapping. This question also deserves a separate study, like in \cite{ZA}. \endremark \demo{Proof of Main Theorem} The key statements are Theorems 5 and 6 above which guarantee the existence of the totally expanded model with {\it no\/} small contour subsystems. In fact, these two theorems opened the possibility of the very {\it formulation\/} of our result based on the construction of the quantities $h_t(y)$ having rather strong and useful properties. Noticing this, one has to add only a few additional observations. 1) If $y$ is a ground state, i.e. it satisfies (3.73), then no contour system $\gb$ such that $(x_{\gb})^{\ext}=y$ is residual. Really, we have from the definition of $A^*_{G}(\square, y)$, $\square =\square(\gb)$ (the subscript $G =\supp \gb$ denotes the $G$ -- expanded model (3.47)) the inequality, with a very small $\tilde \varepsilon$ (of the order $(\varepsilon^*)^{\diam \square}$, see below for $\varepsilon^*$) $$ A^*_G(\square, y) \leq \tilde \varepsilon \diam \square. \tag 3.80 $$ One could take, from (3.73), even {\it zero\/} on the right hand side of this relation if the appropriate {\it totally\/} expanded model would be taken here. So we have to use first an estimate (similar in its nature to (3.42)) $$|A^*_G (\square,y) - A^*_{\text{totally expanded}} (\square,y)| \leq (\varepsilon^*)^{\diam \square} \diam \square $$ where $\varepsilon^* = C \varepsilon$. Thus we see, that the smallness of any contour system $\gb$ with the external colour $y$ satisfying (3.73) is (having in mind the -- just mentioned -- slight difference betweeen different mixed models which could be considered there) automatically fulfilled (actually with a very {\it small\/} constant $\tilde \varepsilon$ instead of $\tau'$ !) and therefore, under such boundary condition $y$, there is {\it no\/} difference between the original \ps\ model and the metastable one. Thus any contour system $\Cal D$ with boundary condition $y$ is small. Existence of such systems clearly implies also an existence of such systems with connected $\vvd$. These systems are, however, extremally small, because they are small and all their subsystems are residual. \footnote{More precisely, nonrecolorability of subsystems of $\Cal D$ is guaranteed in a slightly different mixed model (than the totally expanded one) only. This means that the constants like $\varepsilon'$ in the proof of Theorem 6 have to be slightly changed and our choice of $\tau'$ must be slightly more careful than we argued before.} By Theorem 6, they are recolorable and this is a contradiction with the definition of the fully expanded model. 2) On the other hand, if $y$ is {\it not\/} a ground state then we will show that some residual systems having the external colour $y$ {\it do exist\/}: Namely, then there is some $\tilde y \in \es$ differing from $y$ only on some layer $L$ of a finite width, say $d$, such that the vertical sum below (it does not depend on $t$) is $$ \sum_{t'\in[t]}h_{t'}(y)-h_{t'}(\tilde y) \geq \delta \tag 3.82 $$ for a suitable $\delta >0$ . Take a very large box $B\subset \zw$ such that $$ \tau |\partial B| << \delta |B| \tag 3.83 $$ and consider the volume, having the form of a ``desk'' $ \Lambda = \{t\in\tilde L, \hat t \in B\} $ where $\hat t$ denotes the projection to $\zw$ and $\tilde L$ is another, thicker layer containing the above layer $L$ ``sufficiently inside itself''. The thickness $2d$ would be amply sufficient; the purpose of the thicker layer is to control all the nonnegligible (with respect to $\delta$) cluster terms acting in the difference $\sum (h_t(y) -h_t(\tilde y))$. Take the configuration $x=y$ outside of $\Lambda$, $x=\tilde y$ inside $\Lambda$. Then, if we compute the quantity $A^*(\Lambda)$ for the volume $ \Lambda $, we have according to (3.47) the folowing bound from below: $$ A^*(\Lambda) = \sum_{t\in\Lambda}(h_t(y)-h_t(\tilde y)) \geq (\delta -\epsilon)|B| \gg \tau |\partial B| \ \ \text{i.e.} \ \gg \tau|\diam \Lambda| , \tag 3.84 $$ which shows that $\Lambda$ is not a small volume i.e. some residual contours do exist ! This argument is absolutely straightforward for the original spin model (2.2) where contours of almost arbitrary shape can be found. For an abstract \ps \ model the above conclusion (on the existence of residual contours under unstable boundary conditions) requires in fact the assumption that contours having arbitrarily big diameters really {\it do exist\/}, in the abstract model, for any external colour $y \in \es$. \footnote{ This is of course fulfilled for abstract \ps \ models arising from (2.2). One could define artificial models {\it not\/} allowing such big contours; then a ``one sided'' modification of the statement of Main Theorem must be made.} 3) Moreover one has, under stable boundary conditions $y$, quickly converging expansions of partition functions in any volume, and this implies the validity, as we explain below in (3.93), of the properties of the ``$y$ -- th Gibbs state''$P^y$ stated in the Main Theorem. \enddemo \definition{Generalization to quickly decaying infinite range Hamiltonians. An outline} \enddefinition In Corollary, end of section II we formulated the concept of an essentially one dimensional interaction and outlined how to work with it. This concept is just a realization of the idea to ``pack'' all the interactions $\Phi_A$ resp. cluster expansion terms (the latter can be interpreted as some additional interactions, too) into corresponding (as small as possible) cubes. It is important to assume a quick decay of the long range interactions. Otherwise there would be problems with establishing of the Peierls condition for the ``aggregates'' defined below. If we inspect all the technical steps we made during our development of the expansion process in part III then the conclusion is that there are the following two groups of problems caused by additional infinite range (sufficiently quickly decaying) interactions : 1) Let us stick first to the formulation given in terms of the abstract \ps \ model (as always up to now; below we will briefly mention also the case of spin models developed in \cite{EMZ}).) Notice that instead of an abstract \ps \ model we can already start with a nontrivial {\it mixed model\/} as well. Actually, our Theorems 3 and 4 and all the treatment of the sections 2 - 7 is already given in a form prepared for such a generalization -- assuming that the original, long range, essentially one dimensional interactions fit our assumptions (3.1) on the given mixed model. We have to explain what (3.1) would mean for essentially one dimensional interactions $\Phi_{\square}$: Define $\conn \square$ as\ $C \log \diam \square$\ for a suitable $C > 1$; then apparently only a decay with a sufficiently big power of distance is required, so that the machinery of sections 2 -- 7 could be applied without a change. \newline 2) Few comments about the {\it spin\/} models (used in the applications made in \cite{EMZ}): If we have a nontrivial additional long range interaction $H^{\omega,\varepsilon}$ in the situation of the short range model (2.2) then the following auxiliary construction has to be made, to adapt the spin case to the case of abstract \ps \ models studied so far. Given the perturbative interactions $\Phi_{\square}^{\omega,\varepsilon}(x)$ which now act also {\it inside\/} of the contours (assuming that the restriction of a given configuration to a cube is stratified) compute the quantities $e_t(y)$ as in (2.8), taking in account also the {\it long range interactions\/} $\Phi_{\square}$. Of course, we cannot adjust the contour energies $E(\gb)$ to fulfil (2.12) exactly in the case of infinite interactions. Therefore, let us try to express the necessary corrections as additional (weak!) {\it interactions acting on cubes intersecting contours\/} only: Define $$\Phi^{\text{corr}}_{\square}(x_{\square}) =\sum_y \xi_{\square}^{x,y} \Phi_{\square}^ {\varepsilon,\omega} (y_{\square}) \tag 3.85$$ for any such cube $\square$, where the sum is over all stratified $y$' s which are equal to $x$ at some stratified point of $x$ belonging to $\square$. (Typically, the sum contains only {\it one\/} term.) The coefficient $ \xi < 1$ is computed such that the expression (2.12) would be {\it exact\/} with the addition of the interactions $\Phi^{\text{corr}}_{\square}(x_{\square})$. (This is clearly possible, we omit formulas analogous to (2.11) for $\Phi^{\text{corr}}_{\square}(x_{\square})$.) These correction potentials can now be treated by the following ``high temperature expansion'': Given a configuration $x$ take the product over the family $\Cal C(x)$ of all cubes intersecting, but not included in, the union $B$ of supports of all contours of $x$. (Contours are still defined with respect to the ``unperturbed'', short range Hamiltonian $H_0$!) $$ \prod_{\square \in \Cal C(x)} \exp(-\Phi_{\square}^{\text{corr}} (x_{\square})) = \prod_{\square \in \Cal C(x)} (1+ k_{\square}^{\text{corr}} (x_{\square})) = \sum_{\Cal S} \prod_{\square \in \Cal S} k_{\square}^{ \text{corr}} (x_{\square})) \tag 3.86$$ where the sum is over all subsystems $\Cal S \subset \Cal C(x)$ of the collection of cubes above and where $ k_{\square}^{\text{corr}} = \exp(-\Phi_{\square}^{\text{ corr}})- 1$ decay like $\Phi_{\square}$. Write $ k_{\square}^{\text{corr}}(x_{\square})$ also in the exponential form $$ k_{\square}^{\text{corr}} (x_{\square}) = \exp(-\Phi_{\square}^{\text{new}}(x_{\square})) $$ and perform the corresponding expansion of the right hand side of the expression $$\prod_{\square \in \Cal C(x)} \exp(-\Phi_{\square}^{\text{corr}} (x_{\square})) \prod_{\{\gb_i\}}\exp(- H_{\text{ref}} (\{\gb_i\})) \tag 3.87 $$ where $H_{\text{ref}}$ denotes the ``reference'' short range hamiltonian (2.12) but with the {\it long range interactions incorporated into the definition of\/} $e_t(y) $ (not $E(\gb)$ !) as mentioned above. Then a new abstract \ps \ model with the same densities $e_t(y)$ as above is obtained, if we define the {\it new contours\/} of the {\it new abstract \ps \ model\/} as {\it conglomerates\/} of the original contours and ``high temperature cubes'' \ $\square$ \ from the product above. The new conglomerates play the role of usual contours of the setting (2.12) and their energy is the sum of the Peierls energies of original contours contained in the conglomerate and of the energies $\Phi^{\text{new}}_{\square} (x_{\square})$. We may now check the Peierls condition of the type (2.14), interpreting cubes of diameter $n$ as suitable commensurate collections of $C \log n$ points (in order to minimize the entropy estimates for the conglomerates produced by (3.87)) for the aggregates of the cubes from the above expansion and contours of the configuration $\{\gb_i\}$. We do not study this case in a detail yet but notice that an exponential decay of the type (2.28) is sufficient even for $\omega$ close to 1 if $\varepsilon$ in (2.28) is sufficiently small. We mention that the case of infinite range interactions in the usual translation invariant \ps \ theory was first systematically treated by papers \cite{YMP}. \vskip1mm \head 9. Properties of Typical Configurations, Gibbs States in finite Volumes \endhead We are still in the situation of an abstract Pirogov -- Sinai model (and back in the case of finite range Hamiltonians). Given $y \in \es$ denote by $X^y_{\text{meta}}$ the configuration space of all configurations of the $y$ -th metastable model. More precisely denote by $X^y_{\text{meta}}(\Lambda)$ the configuration space of all configurations $x_{\Lambda}=(x_{\Lambda}^{\text{best}}, \Cal D) $ such that $y $ is the external colour of $x_{\Lambda}^{\text{best}}$ and $\Cal D$ is strictly diluted in $\Lambda$ i.e. $\Cal D \ssubset \Lambda$. We have the probability measure $P^y_{\text{meta},\Lambda}$ on $X^y_{\text{meta}}(\Lambda)$, the corresponding partition function $Z_{\text{meta}}(\Lambda,y)$ being given by summation of $\exp(-H(x_{\Lambda}|y))$ over all $x_{\Lambda} \in X^y_{\text{meta}}$. The Gibbs factor $H(x_{\Lambda}|y)$ is given by (2.12). Write $P_{\Lambda}^{y}$ instead of $P^y_{\text{meta},\Lambda}$ if $y$ is stable. Having defined these ``strictly diluted'' Gibbs conditioned probabilities $P^y_{\text{meta},\Lambda}$ the question now is whether some limit $$ P^y_{\text{meta}}(\Cal E ) = \lim_{\Lambda \to \zv} P_{\text{meta},\Lambda}^y(\Cal E) \tag 3.88 $$ exists for a sufficiently rich collection of events $\Cal E$. ``Sufficiently rich'' would mean, in the {\it spin\/} model, e.g. the collection of all cylindrical events i.e. events depending on a finite number of coordinates. In the abstract \ps\ model, a simplest example of such an event is $\Cal E(t)$ below in (3.89). The limit then gives for $y$ {\it stable\/}, a prescription for a {\it Gibbs measure\/} $P^y$ on the whole configuration space $\ex$. (More precisely prescription for the probabilities of some special events like below. We omit here the full proof of the fact that $P^y$ given by formulas below really gives a uniquely defined $\sigma$ -- additive probability measure.) For nonstable $y$ this limit can be interpreted as the ``metastable Gibbs measure'', denoted by $P^y_{\text{meta}}$. Let us show for example that the limit $$ P^y(\Cal E (t)) = \lim_{\Lambda \to \zv} P_{\Lambda}^{y}(\Cal E(t)) \tag 3.89 $$ exists for any stratified $y$ and for any $t \in \zv$ if the event $\Cal E(t)$ is defined as follows: \ ``$ x \in \Cal E(t) \ \ \text{iff} \ \ t \notin \vvd$ where $\Cal D$ is the collection of all contours of the configuration $x$''. (For nonstable $y$ we have an analogous statement for $P^y_{\text{meta}}$.) This event will be called ``$t$ is strictly {\it exterior\/} point of the configuration $x$''. Notice that for each $\Lambda$ we have the formula (and analogously in a general metastable situation) $$P_{\Lambda}^y(\Cal E(t)) = (Z^y_{\updownarrow}(\Lambda))^{-1} Z^y_{\updownarrow} (\Lambda \setminus t) \exp(-e_t(y)). \tag 3.90 $$ We can expand, in the totally expanded model, both the partition functions on the right hand side of (3.90). The possibility of such an expansion follows from the very notion of stability of $y$. We have the following expression (analogous expansions will be commented below also for some more general boundary conditions): $$ \log P^y_{\Lambda}(\Cal E(t)) = \sum_{T : \ T \cap \cap (\Lambda^c \cup \{t\}) = \{t\}} k_T(y) \tag 3.91 $$ where the quantities $k_T(y)$ are from the expansion (3.1) of $Z^y_{\updownarrow}(\Lambda)$ resp. of $Z^y_{\updownarrow}(\Lambda\setminus t)$ and the sum is over those (quickly decaying !) quantities $k_T$ only which ``touch'' $t$ in the sense that $T$ does {\it not\/} satisfy the condition $T \ssubset \Lambda \setminus t$ \ but does satisfy the condition $T \ssubset \Lambda$. Clearly, then we have ($\varepsilon$ is from (3.1) and $C =C(\nu)$) an approximate relation $$ 1 - P^y_{\Lambda}(\Cal E(t)) \ \leq C \ \varepsilon . \tag 3.92 $$ This suggests that the ``islands'' of almost any configuration $x_{\Lambda}= (x_{\Lambda}^{\text{best}},\Cal D)$ (see below) are really mostly ``small sized and rare'' because they do not usually intersect a given (arbitrarily chosen) point $t \in \zv$. To make this intuitive description of almost any configuration (which is quite characteristic for the Pirogov -- Sinai theory and for the phase picture this theory gives) more technically more accessible, we will define below also the ``frames'' of islands $(x,\Cal D)$ in a way grasping also some important features (namely the appearance of residual contours) of the regime appearing {\it inside of $\vvd$\/} and thus allowing also {\it expansion formulas\/} for the event ``a given frame appears''. \definition{External contours, islands} Define the relation $ \gb \to \gb' $ whenever ($ \supp \gb \prec \supp \gb'$ and moreover) $\vv \subset V_{\updownarrow}(\gb') $. (We consider {\it no shifts\/} of $\gb'$ here!) Let us interpret the relation $\gb \to \gb'$ by saying that contour $\gb$ is ``inside'' of the contour $\gb'$. Say \footnote{One should not take this characterization too literally. Of course the usual intuitive concept of ``exteriority'' fits the scheme above but also some contours $\gb$ which are ``mostly outside'' of $\gb'$ satisfy the condition above. However, in contrary to the concept of admissibility which will be {\it fundamental\/} in what follows the relation ``$\to$''will {\it not\/} be employed below except of some notes at the very {\it end\/} of the paper, when explaining the meaning of our Main Theorem.} that $\gb$ is an external contour of an finite contour system $\Cal D$ if $\gb \to \gb'$ for no contour $\gb'$ of $\Cal D$. Clearly, $ \cup \Sb \gb \in \Cal D \endSb \vv = \cup \Sb \gb \in \Cal D^{\text{ext}} \endSb \vv $ where the second union is over all external contours of $\Cal D$. Any connected component $I$ of $\vvd$ (equivalently, one can take $\Cal D^{\text{ext}}$ here) will be called an {\it island\/} of $\Cal D$. Denote by $\Cal D_{I}= \{\gb \in \Cal D: \vv \subset I\}$, analogously define $\Cal D^{\text{ext}}_{I}$. \footnote{ Maybe ``archipelago of contours'' would be an appropriate name. Typically, however, islands are mutually external, simply connected sets and for islands with ``holes'', the external colour of the island penetrates also inside the holes (Proposition below). So, {\it any\/} island is effectively an ``external one''.}\enddefinition \proclaim{Proposition} For each island $I$ of any configuration, one can determine the ``colour'' $y = x_I= x_{\Cal D_I} \in \es $ of the island (i.e. of the system $ \Cal D_I$) which appears also inside of the (possible) interior components of $(V_{\updownarrow}( \Cal D_I))^c$. Any contour of $\Cal D$ either belongs to $\Cal D_I$ or is isolated from it.\endproclaim \demo{Proof}This is a geometrical statement saying that regions with different ``interior'' colours (which can appear ``inside'' of the contours of the island $I$) are already covered by $V_{\updownarrow}(\Cal D_I)$. This is clear if $\Cal D_I$ is a single contour (then obviously $V(\Cal D) \subset V_{\updownarrow}(\Cal D)$). Otherwise some additional considerations are needed, we omit them here. \enddemo \definition{Frames of a configuration, the probability $P^y_{\Lambda}[\gb]$} A contour subsystem $\gb \subset \Cal D; $ of some island $\Cal D_I$ of $\Cal D$ will be called a {\it frame\/}, more precisely frame of the island $\Cal D_I$), if it contains all the external contours of the system $\Cal D_I$ and moreover the system $\Cal D_I \setminus \gb$ is strongly diluted in $(\supp \gb)^c$ and {\it metastable\/}. (In other words, frame of $\Cal D_I$ is formed by $\Cal D_I^{\text{ext}}$ and the {\it residuum\/} of $\Cal D_I \setminus \Cal D^{\text{ext}}_{I}$.) Denote by $P^y_{\Lambda}[\gb]$ the probability (in $P^y_{\Lambda}$; below we are mainly interested in the stable values of $x_{\gb}^{\text{ext}}= y$ and in the thermodynamic limit $\Lambda \to \zv$) of the event ``$\gb$ is a frame of a given configuration''. The following theorem is then an analogy of the fundamental relation (3.23). \enddefinition \remark{Note} Above, metastability is meant in the local colours induced by $\gb$ outside of $\supp \gb$. We are able to decide whether $\gb' \subset \Cal D \setminus \gb$ is recolorable or metastable, because the covering cube of $\gb'$ is still contained in $(\supp \gb)^c$. The case of a ``small'' system $\gb$ such that $\vv \setminus \supp \gb$ does not contain other contours and everything what can happen outside of $\vv$ is (meta)stable is the most characteristic one, of course, and $\gb$ is commonly either a single contour or a collection of one external contour and some interior residual ones. \endremark \proclaim{Theorem 8. Expression for $P^y_{\Lambda}[\gb]$ } We have the formula, for any contour system $\gb \ssubset \Lambda$ (this is a more detailed version of (2.24); \ $y$ denotes the external colour of $\gb$) $$ P^y_{\Lambda}[\gb] = \exp (-F_{\text{full}}(\gb)) \ \exp(\sum_{ T\cap \cap \Lambda^c \ne \emptyset}^{T \cap \cap \supp \gb \ne \emptyset} k_T^{\text{cor}}(\gb) ) \tag 3.93 $$ where $F_{\text{full}}(\gb) = F_{\text{full}}^{\infty}(\gb)$ is like in the definition (3.22), the superscript $\infty$ indicating that we take the totally expanded, not the $G$--expanded, $G =\supp \gb$ (like in Theorem 3) model when constructing the quantity $F^{\infty}_{\text{full}}(\gb) = E(\gb)-A^{\infty}_{\text{full}}(\gb) $. \footnote{In contrary to Theorem 3, we {\it do not want \/} to expand here the frame $\gb$. We do not even assume a Peierls type inequality for $F_{\text{full}}(\gb)$. However, then it can happen (for nonstable colour $x_{\gb}^{\text{ext}}$ only !) that the statement (3.94) below is violated. (The difference between $F^{\infty}$ and $F^{G}$ plays almost no role in it.)} The quantities $k_T^{\text{cor}}(\gb)= k_T(x_{\gb}) -k_T(x_{\gb}^{\text{ext}})$ represent, in the notations analogous to that of (3.23), just the difference $F_{\text{full}}^{\infty}(\gb) - F^{\infty}_{ \Lambda}(\gb)$; the sum over $T$ being empty in the thermodynamic limit. \footnote{ A formula similar to (3.93) (and closer to the situation of (3.23)) could be obtained when taking \ $F_{\text{full}}^G (\gb), \ G =\supp \gb$ there. However, then it would describe the {\it conditional probability\/} that the frame $\gb$ appears under a condition that all the bigger contour systems $\Cal D, \ G \prec \supp \Cal D$ are forbidden ! } \endproclaim \remark{Notes} 1. Notice that $F_{\infty}$ is a horizontally translation invariant (nonlocal !) quantity.\newline 2. The concept of a frame is our true substitution for the usual concept of an external contour (the latter being {\it not\/} so important here). Notice that ``to be an frame'' is {\it not\/} a cylindrical event. However, this is an event ``close'' to a cylindrical one and it can be expressed much more nicely, by the formula (3.93), than the purely cylindrical events. Also, the purely cylindrical events like $ \Cal E_{\gb} =\{ \Cal D: \gb \subset \Cal D\}$ can be expressed in terms of ``frame events'', by specifying the very frame $\gb' \subset \Cal D$ to which $\gb$ either {\it belongs\/} or (we consider here, for simplicity, only these two ``extremal'' situations) which {\it controls\/} $\gb$ in the sense that $\vv \subset V_{\updownarrow}(\gb')$. In both cases we have cluster expansion formulas based on (3.93) for these events; more direct formulas (just (3.93)) are obtained in the former case. \footnote{ The probability of $\Cal E_{\gb}$ is then the sum of the probabilities of different (``connected'', by Theorem 6) frames $\gb', \gb' \owns \gb$ -- for which we have (3.93) and, therefore, for stable $y$, also (3.94)!} In the latter case, we have the formula (3.93) for the probability of any frame $\gb'$ ``controlling'' $\gb$ and {\it then\/} we have usual formulas for the {\it conditional probability\/} of the appearance of $\gb$ under $\gb'$. \footnote{ We do not formulate here these conditional probabilities (of the more detailed events happening in the ``interior'' $V_{\updownarrow}(\gb') \setminus \supp \gb' $ of some frame $\gb'$). This is already a straightforward task, using the properties of conditional Gibbs distributions.} \endremark \demo{An outline of the proof of Theorem 8} This is just another application of the method of (3.23) (more precisely (3.25)), for $\Cal D = \emptyset$. Expand (by suitable modification of Main Theorem applied only to the volume $(\supp \gb)^c$ under a condition $\gb$ (not to the whole lattice $\zv$!) the partition function of the event ``$\gb$ is a frame, with external colour $y$, of a configuration $\Cal D$ in $\Lambda$'' and divide it by the expansion of the whole partition function $Z_{\updownarrow}(\Lambda)$. Then (3.93) is obtained, with a value of $F_{\text{full}}(\gb)$, which is {\it very\/} close to the value $F(\gb)$ introduced in section 3. The difference from Theorem 3 is that here we do {\it not\/} recolor $\gb$ (even if it could be recolored at the moment when the recoloring proces ``reaches the level of $\supp \gb$'') and then continue the recoloring process only in the set $\zv \setminus \supp \gb$. \footnote {Instead of recoloring in the whole $\zv$. Namely, the method of Theorems 3,4,5 can be extended to any, even irregular, ``lattice'' if the statement on horizontal invariance of the result is rephrased properly. The possibility to use boundary conditions which are only locally from $\es$ was yet footnoted in Theorem 3.} One uses an analogous (but not exactly the same) arguments as those used there. Recall that the totally expanded model used in the definition of $F^{\infty}(\gb)$ employs also clusters containing contours ``bigger than $\gb$'' (but isolated from it). In fact, the difference between the values of $F^G(\gb), G=\supp \gb$ (used in section 3) and $F^{\infty}(\gb)$ (used here) is of the order $ \varepsilon^{\diam \gb}$ only. It is given by contributions $k_T$ of clusters $T$ containing elements bigger (in $\prec$) than $\supp \gb$. \enddemo \proclaim{Corollary of Theorem 8} If a frame $\gb$ is recolorable then $P^y[\gb]$ satisfies the estimate, with $\varepsilon = \exp(-\hat \tau)$ and $\hat \tau$ slightly smaller than $\tau / 12\nu$ , $$ P^y[\gb] \leq \varepsilon^{\conn \gb}\ . \tag 3.94 $$ The probability of the event ``there is a frame around $t$ having a diameter $\leq d$'' can be estimated, for $y$ stable, by $\text{const} \cdot \varepsilon^d$. The mean relative area occupied by the supports of such frames $\gb$ resp. by the volumes $\vv$ of almost any configuration is then equal, in the Van Hove limit, to some $\varepsilon' = \varepsilon'(\varepsilon) $ resp. some $\varepsilon'' = \varepsilon''(\varepsilon) $ independently of the particular choice of the configuration. There is an exponential decay of correlations in the probability $P^y$ and $P^y$ has support in the countable union of compact sets $X^{\text{abstract}}_{\tilde I}$, $\tilde I$ shift of $I$. \endproclaim We do not prove these (rather straightforward once (3.93) was established) facts here. \remark{Note} The formulas like (3.93) only {\it suggest\/} that $P^y$ exists really a {\it probability measure\/} on $\ex$ with the properties stated above. Also, to gain a control over some more general diluted or ``almost diluted'' (see below in last section for natural examples of more general boundary conditions which are less restrictive than the strictly diluted ones) partition function and finally to interpret the measure $P^y$ even as a Gibbs measure on $\ex$ i.e. as a Gibbs measure of the {\it original Hamiltonian\/} (2.2) some additional, more or less straightforward work is needed. See below for some comments. \endremark \definition{Stratified Gibbs States in Arbitrary Finite Volumes} \enddefinition Finally we give here a short information about the properties of Gibbs ensembles in finite volumes arising from some more general boundary conditions than the ``strictly diluted'' stratified ones which we used everywhere before. Our aim here is not to investigate the partition functions under arbitrary boundary conditions in {\it full generality\/} (this question and the related question of the completeness of the phase picture constructed by Main Theorem deserve a separate study) but only to study some noncomplicated and quite natural boundary conditions which are still, in a sense, ``near'' to the requirement of the strict $y$ -- diluteness in $\Lambda$, $y \in \es$ but are formulated in a simpler and more traditional way {\it not\/} based on the notion of tightness (and isolation) developed in this paper. It will be convenient to remain still in the framework of the abstract \ps \ situation. Namely, even the context of (2.2) with a stratified $x_{\Lambda^c} = y_{\Lambda^c},\ y \in \es$ can be described in the language of (2.12), assuming now however that contours of the spin configurations $x_{\Lambda} \cup y_{\Lambda^c}$ {\it may even intersect\/} the set $\Lambda^c$ (but apparently only at the condition that they also intersect $\Lambda$). Denote by $X^{\text{gen}}(\Lambda, y)$ the collection of all such (more general than strictly diluted) configurations of the abstract \ps \ model and by $P^{\text{gen}}_{\Lambda,y}$ the corresponding Gibbs probability on this ensemble. Such a broader concept includes also the traditional partition functions (2.4) if $x_{\Lambda^c}$ equals to some $y_{\Lambda^c},\ y \in \es$. \definition{On some more general, $y$ -- like and ``fuzzy'' boundary conditions}\enddefinition One can generalize the arguments given below even to some more general $y$ -- like boundary conditions on $\partial \Lambda$. We mean the boundary condition $x$ given on $\Lambda^c$ such that $x =y$ holds for ``most'' points of $\partial \Lambda^c$. One should be a little bit careful what does mean the requirement ``for most points of $\partial \Lambda$''. However, we will not discuss literally such a generalization here. In fact, it is more instructive to study, for {\it any\/} volume $\Lambda$, the following ``fuzzy'' boundary conditions given by some {\it additional Hamiltonian\/} acting ``outside of the volume $\Lambda$ but in the vicinity of $\partial \Lambda$'' (say in some layer $\partial_r \Lambda^c$) which ``supresses sufficientlly all the deviations from the horizontally constant $y$ --th regime'' with the help, say, of additional chemical potential ``disfavouring everything else than pure $y$''. It can be shown that such a ``fuzzy'' boundary condition gives the same phase in the thermodynamic limit (for reasonable (conoidal) volumes) as the strictly diluted $y$ -- th boundary condition studied before, if $y$ is stable. Far from $\partial \Lambda$ one will not see any noticeable difference between those boundary conditions. Formulas like (3.93) enable to prove these statements quite easily. \footnote{ More precisely the action of any such boundary condition on some $\partial_r \Lambda^c$ can be modelled with the help of a suitable {\it strictly\/} diluted boundary condition in a {\it suitable larger volume\/} $\tilde\Lambda$ with $\dist(\Lambda,\tilde \Lambda^c) > r$, however with some additional energies (denoted by $\tilde E, \tilde e$) added (for ``fuzzy''boundary conditions) to the original energies (2.12). The added energies are such that $\tilde E(\gb) \geq 0$ and $\tilde e_t(z)\geq 0 ,\ \tilde e_t(y) = 0,\ z \in \es$ and they act only in $\tilde \Lambda \setminus \Lambda$. We could formulate everything in such a more general way already starting from (2.12) if all the additional requirements above could be considered as a ``weak preturbation of pure $y$''. The only change would be that the quantities $k_T(z),\ z \in \es$, lose their horizontal invariance in the case when $T$ touches $\tilde \Lambda \setminus \Lambda$ and also $e_t(z), z \in \es, y \ne z$ are increased in the strip $\tilde \Lambda \setminus \Lambda$. Then, a short look at formulas like (2.26), (3.91), (3.93) shows that the conclusions like the stability of the $y$ -- th phase constructed by these more general boundary conditions in conoidal volumes $\Lambda$ (see below), and also in the infinite volume limit {\it remain valid\/} if we keep $r$ in $\partial_r \Lambda^c$ fixed and not very big (depending on the value $\tau$). The case of boundary conditions which are `'far from pure $y$'' is, however, a more complicated problem.} \definition{On shapes of $\Lambda$ with negligible entropic repulsion/attraction effects} \enddefinition Concerning the shape of the considered volumes $\Lambda$, one has to be a little bit more careful. Namely for volumes $\Lambda$ whose boundary contains huge planar parts (like volumes having a shape of a `` desk'', or even cubes!) the boundary term representing the ``entropic repulsion or attraction'' can be important. In fact one can compute these effects from relations like (3.93) (more thorough study is prepared for the paper \cite{HMZ}) and the stability of $y$ can lose its validity near the boundaries of such volumes, if more general boundary conditions than the strictly diluted ones (which were, among other properties, ``designed not to feel these subtleties'') are considered. Namely, if the configuration in $\Lambda$ is not strictly diluted, $\vv \cap \Lambda^c \ne \emptyset$ then $A^{\infty}_{\text{full}}$ from Theorem 8 can change its value significantly if $V(\gb)$ sticks to the flat part of $\partial\Lambda$. The corresponding generalization of (3.93) then uses the quantity $$ A_{\Lambda}^{\infty}(\gb,x|\vv) = \sum_{t\in V(\gb)}(e_t(y)- e_t(x)) + \sum\Sb T\ssubset \vv\setminus \supp\gb \endSb ^{T \ssubset \Lambda} k_T(x)- \sum_{T\ssubset \vv}^{T\ssubset \Lambda}k_T(y) \tag 3.95 $$ (where $x = x_{\gb}$) in the fully expanded model and this can be written as (compare (2.24)!) $$ A_{\Lambda}^{\infty}(\gb,x|\vv) = \sum_{t\in V(\gb)}(h_t(y)- h_t(x)) + \Delta(\gb) \tag 3.96 $$ where $\Delta(\gb)$ (we have to check that this quantity ``does not erode'' $E(\gb)$) is given by $$ \sum_{t\in V(\gb)}(e_t(y)-h_t(y)- e_t(x) +h_t(x)) + \sum^{T\ssubset \Lambda}\Sb T\ssubset \vv \setminus \supp \gb \endSb k_T(x)- \sum_{T\ssubset \vv}^{T\ssubset \Lambda}k_T(y). \tag 3.97 $$ This {\it can\/} be a dangerous quantity if the ``upper (lower) ceiling of $\Lambda$ is very close to the upper (lower) ceiling of $V(\gb)$, because then the last two sums give a term proportionate to the area of the flat part of $\vv \cap \Lambda$ ! \footnote{ This cannot happen in conoidal volumes where the last sum is of the order $\supp \gb$ only.} and in such a way, the stability of $y$ can be {\it destroyed\/} (or, on the contrary, established for some {\it nonstable\/} $y$) in these volumes which are ``very flat from below resp. from above''. Then, some residual frames $\gb$ marking the shift to some other, more stable (in $\Lambda$ !) horizontally invariant regime $\tilde y$ can appear (only in {\it huge\/} volumes $\Lambda$ and only near their boundary). In order to control these effects we have to introduce some {\it additional\/} (rather mild) assumptions on the ``shape'' of $\Lambda$, like that there are ``no flat ceelings in $\partial \Lambda$''. (This will assure that $A^{\infty}_{\Lambda}(\gb)$ is rather benign, not eroding $E(\gb)$). The following definition gives a characteristic example of such a volume. \definition{Conoidal volumes} Say that a volume $ \Lambda$ is a conoidal volume (or conoid) if it contains, with each ``horizontal set'' $ B \subset \zv_m \cap \Lambda$, where $\zv_m$ is the collection of points of $\zv$ with the fixed last coordinate $t_{\nu}=m$, also the whole ``cone'' $\{t \in \zv : \dist(t,B) \leq C_{\dist(t, \partial B)}\}$ where $ C \log r < C_r < r$ holds with not too small constant $C$. \footnote{ The name ``cone'' is appropriate, of course, only if $C_r \approx r$ (like $C_r =r/2$). $\partial B$ is taken in $\zv_m$.} \enddefinition Now, in conoidal volumes \footnote{ So, parallelpipeds are slightly awkward volumes in principle. One has to be a little bit careful what stability means, especially near their boundaries, and this is a rather subtle question. Also, some ``almost stable'' $y \in \es$ can look like stable in suitable large parallelopipeds. 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