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\let\P=\Pi \let\Ps=\Psi \let\Si=\Sigma \let\X=\Xi \let\Y=\Upsilon %%%%%%%%%%%%%%% DEFINIZIONI LOCALI \let\ciao=\bye \def\fiat{{}} \def\pagina{{\vfill\eject}} \def\\{\noindent} \def\bra#1{{\langle#1|}} \def\ket#1{{|#1\rangle}} \def\media#1{{\langle#1\rangle}} \def\ie{\hbox{\it i.e.\ }} \let\ig=\int \let\io=\infty \let\i=\infty \let\dpr=\partial \def\V#1{\vec#1} \def\Dp{\V\dpr} \def\tende#1{\vtop{\ialign{##\crcr\rightarrowfill\crcr \noalign{\kern-1pt\nointerlineskip} \hskip3.pt${\scriptstyle #1}$\hskip3.pt\crcr}}} \def\otto{{\kern-1.truept\leftarrow\kern-5.truept\to\kern-1.truept}} \def\Z{{\bf Z^d}} \def\supnorm#1{\vert#1\vert_\infty} \def\grad#1#2{(\nabla_{\L_{#1}}#2)^2} %%%%%%%%%%%%%%%%%%%%% Numerazione pagine \def\data{\number\day/\ifcase\month\or gennaio \or febbraio \or marzo \or aprile \or maggio \or giugno \or luglio \or agosto \or settembre \or ottobre \or novembre \or dicembre \fi/\number\year} %%\newcount\tempo %%\tempo=\number\time\divide\tempo by 60} \setbox200\hbox{$\scriptscriptstyle \data $} \newcount\pgn \pgn=1 \def\foglio{\number\numsec:\number\pgn \global\advance\pgn by 1} \def\foglioa{A\number\numsec:\number\pgn \global\advance\pgn by 1} %\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm %\foglio\hss} %\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm %\foglioa\hss} % %%%%%%%%%%%%%%%%% EQUAZIONI CON NOMI SIMBOLICI %%% %%% per assegnare un nome simbolico ad una equazione basta %%% scrivere \Eq(...) o, in \eqalignno, \eq(...) o, %%% nelle appendici, \Eqa(...) o \eqa(...): %%% dentro le parentesi e al posto dei ... %%% si puo' scrivere qualsiasi commento; %%% per assegnare un nome simbolico ad una figura, basta scrivere %%% \geq(...); per avere i nomi %%% simbolici segnati a sinistra delle formule e delle figure si deve %%% dichiarare il documento come bozza, iniziando il testo con %%% \BOZZA. Sinonimi \Eq,\EQ,\EQS; \eq,\eqs; \Eqa,\Eqas;\eqa,\eqas. %%% All' inizio di ogni paragrafo si devono definire il %%% numero del paragrafo e della prima formula dichiarando %%% \numsec=... \numfor=... (brevetto Eckmannn); all'inizio del lavoro %%% bisogna porre \numfig=1 (il numero delle figure non contiene la sezione. %%% Si possono citare formule o figure seguenti; le corrispondenze fra nomi %%% simbolici e numeri effettivi sono memorizzate nel file \jobname.aux, che %%% viene letto all'inizio, se gia' presente. E' possibile citare anche %%% formule o figure che appaiono in altri file, purche' sia presente il %%% corrispondente file .aux; basta includere all'inizio l'istruzione %%% \include{nomefile} %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \global\newcount\numsec\global\newcount\numfor \global\newcount\numfig \gdef\profonditastruttura{\dp\strutbox} \def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax} \def\SIA #1,#2,#3 {\senondefinito{#1#2} \expandafter\xdef\csname #1#2\endcsname{#3} \else \write16{???? ma #1,#2 e' gia' stato definito !!!!} \fi} \def\etichetta(#1){(\veroparagrafo.\veraformula) \SIA e,#1,(\veroparagrafo.\veraformula) \global\advance\numfor by 1 \write15{\string\FU (#1){\equ(#1)}} \write16{ EQ \equ(#1) == #1 }} \def \FU(#1)#2{\SIA fu,#1,#2 } \def\etichettaa(#1){(A\veroparagrafo.\veraformula) \SIA e,#1,(A\veroparagrafo.\veraformula) \global\advance\numfor by 1 \write15{\string\FU (#1){\equ(#1)}} \write16{ EQ \equ(#1) == #1 }} \def\getichetta(#1){Fig. \verafigura \SIA e,#1,{\verafigura} \global\advance\numfig by 1 \write15{\string\FU (#1){\equ(#1)}} \write16{ Fig. \equ(#1) ha simbolo #1 }} \newdimen\gwidth \def\BOZZA{ \def\alato(##1){ {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\hsize\kern-1.2truecm{$\scriptstyle##1$}}}}} \def\galato(##1){ \gwidth=\hsize \divide\gwidth by 2 {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\gwidth\kern-1.2truecm{$\scriptstyle##1$}}}}} } \def\alato(#1){} \def\galato(#1){} \def\veroparagrafo{\number\numsec}\def\veraformula{\number\numfor} \def\verafigura{\number\numfig} %\def\geq(#1){\getichetta(#1)\galato(#1)} \def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}} \def\eq(#1){\etichetta(#1)\alato(#1)} \def\Eqa(#1){\eqno{\etichettaa(#1)\alato(#1)}} \def\eqa(#1){\etichettaa(#1)\alato(#1)} \def\eqv(#1){\senondefinito{fu#1}$\clubsuit$#1\else\csname fu#1\endcsname\fi} \def\equ(#1){\senondefinito{e#1}eqv(#1)\else\csname e#1\endcsname\fi} \let\EQS=\Eq\let\EQ=\Eq \let\eqs=\eq \let\Eqas=\Eqa \let\eqas=\eqa %%%%%%%%%%%%%%%%%% Numerazione verso il futuro ed eventuali paragrafi %%%%%%% precedenti non inseriti nel file da compilare \def\include#1{ \openin13=#1.aux \ifeof13 \relax \else \input #1.aux \closein13 \fi} \openin14=\jobname.aux \ifeof14 \relax \else \input \jobname.aux \closein14 \fi \openout15=\jobname.aux %%%%%%%%%%%%%%%%%%%%%%%%%%%% %\BOZZA \footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm \foglio\hss} \def\refj#1#2#3#4#5#6#7{\parindent 2.2em \item{[{\bf #1}]}{\rm #2,} {\it #3\/} {\rm #4} {\bf #5} {\rm #6} {(\rm #7)}} \numsec=0\numfor=1 \tolerance=10000 \font\ttlfnt=cmcsc10 scaled 1200 %small caps \font\bit=cmbxti10 %bold italic text mode % Author. Initials then last name in upper and lower case % Point after initials % \def\author#1 {\vskip 18pt\tolerance=10000 \noindent\centerline{\ttlfnt#1}\vskip 1cm} % % Address % \def\address#1 {\vskip 4pt\tolerance=10000 \noindent #1\vskip 0.5cm} % % Abstract % \def\abstract#1 { \noindent{\bf Abstract.\ }#1\par} % %\vskip 1cm \centerline{\ttlfnt Instability of Renormalization-Group Pathologies under decimation}\vskip 0.5cm \author{F. Martinelli $^{\dag}$ E. Olivieri $^{\ddag}$} \address{\ninerm \dag Dipartimento di Matematica, III Universit\`a di Roma, Italy \hfill\break{\ddag Dipartimento di Matematica, II Universit\`a di Roma Tor Vergata, Italy } \hfill\break{ e-mail: martin@mat.uniroma.it \hskip 0.5cm olivieri@mat.utovrm.it}} \abstract{\ninerm We investigate the stability and instability of pathologies of renormalization group transformations for lattice spin systems under decimation. In particular we show that, even if the original renormalization group transformation gives rise to a non Gibbsian measure, Gibbsiannes may be restored by applying an extra decimation trasformation. This fact is illustrated in details for the block spin transformation applied to the Ising model. We also discuss the case of another non-Gibbsian measure with nicely decaying correlations functions which remains non-Gibbsian after arbitrary decimation. } \vskip 1cm\noindent {\eightrm Work partially supported by grant SC1-CT91-0695 of the Commission of European Communities} \par \bigskip \bigskip {\bf Key words:} Renormalization-group, Decimation, Non-Gibbsianness, Ising Model \vfill \eject \bigskip \numsec=1\numfor=1 \centerline {\bf Introduction} \vskip 1cm In this note we discuss some aspects of the problem of defining, on rigorous grounds, a renormalization group transformation (RGT) for the Gibbs measure of lattice spin systems of statistical mechanics. For simplicity and without a true loss of generality (see the end of section2), we confine our attention to the average block spin transformation for the 2D Ising model at low temperature and large positive external field. \par In the basic reference [EFS], the authors discuss in a very complete and clear way the possible pathologies that may arise when applying a RGT to a perfectly well behaved Gibbs measure like the one above. To be more specific, suppose that $\mu$ denotes the starting Gibbs measure on a probability space $\O$ and that $$\nu \;=\; T_b\mu\Eq(0.1)$$ denotes the transformed measure, obtained by applying to $\mu$ a RGT $T_b$ acting on "scale" $b$ and defined on a new probability space $\O '$. The system described by the measure $\mu$ and with configurational variables with values in $\O$ is called the "object" system, whereas the system described by $\nu$ with configurational variables with values in $\O '$ is called the "image" system.\par As the authors of [EFS] point out, the main (rather surprising) pathology of the above RGT is that the renormalized measure $\nu$ can very well be {\it non-Gibbsian}, that is the associated system of conditional probabilities is not compatible with any finite norm potential. That may happen even if the starting measure $\mu$, e.g the unique Gibbs measure of some finite range interaction, has all the nice properties describing the one phase region: analitycity, exponential decay of correlations, convergent cluster expansion etc. \par In [EFS] one can find many examples of such pathology. Moreover, the same authors show that the typical mechanism behind the {\it non-Gibbsiannes} of the measure $\nu$ is the appearence of long range order, that is a phase transition, in the object system {\it conditioned} to some particular configurations of the image system. Such long range order implies, in particular , that the measure $\nu$ violates a necessary condition for being Gibbsian, namely "quasi-locality " of its family of conditional probabilities $\{\p_\L\}_{\L}$ . Such a condition, introduced by Kozlov (see [K] and th. 2.12 in [EFS]), roughly speaking implies some sort of uniform continuity of the conditional probabilities $\p_\L$ with respect to the conditioning configuration (see also [BMO] and [FP] for a critical discussion ). \par An interesting example given in [EFS] of the above phenomenon refers to the "decimation trasformation" on scale "b", $T_b^d$, applied to the Gibbs measure $\mu_{\beta, h}$ of the Ising model at low temperature, $\beta>>1$, and small magnetic field $h$. Such a transformation associates to the original measure $\mu_{\beta, h}$ its marginal (or relativization) on the spin variables sitting on the sites of the sublattice $\Z (b)$ of $\Z$ with spacing $b$. In other words one integrates out all the variables in $\Z\setminus \Z (b)$.\par In [EFS] it was proved that, for any given $b$ and for suitable values of $\beta$ and $h$, the measure $$\nu \;=\; T_b^d\mu_{\beta, h}\Eq(0.2)$$ is {\it non-Gibbsian}.\par This fact is the consequence of the degeneracy of the ground state of the Ising model restricted to $\Z\setminus \Z (b)$ if the conditioning spins at the sites of $\Z (b)$ are held fixed in some suitable, particular configuration, for instance all the spins equal to -1, and the value of the magnetic field $h$ is suitably chosen as a function of the lattice spacing $b$. Such a degeneracy, using the theory of Pirogov and Sinai, leads to a first order phase transition at low enough tenperature for the same constrained system.\par One may say that the above "spurious" phase transition comes from the fact that, on a too short length scale $b$ (with respect to the thermodynamic parameters and mainly to $h$), the system is reminiscent of the phase transition taking place at $h=0$. It thus appears plausible that the above pathology could be eliminated and therefore Gibbsiannes recovered, by choosing a large enough spacing $b$ for given fixed values of $\beta$ and $h$; in particular, it should be sufficient to iterate a sufficiently large number of times the same trasformation in order to come back to the space of Gibbsian measures. This is what we actually proved in [MO] together with some additional results like convergence of a cluster expansion for $\nu$ and the convergence of $(T_b^d)^n \mu_{\beta, h}$ to a trivial fixed point as $n\to \infty$.\par In [EFS] there is another, more subtle, example of pathology, referring to the so called "block averaging transformation" for the 2D Ising model. It is this example and the associated pathology the main object of the present note.\par The trasformation is defined as follows. Suppose to partition the lattice ${\bf Z^2}$ into 2x2 blocks $Q_i$ and let us denote by $m_i$ one of the five possible values of the magnetization (sum of the spins) inside the block $Q_i$. Then the transformed measure $$\mu^B(\{m_i\})\;=\;T_2^B\mu_{\beta ,h}$$ is defined simply as the probability distribution of the variables $m\,\equiv\,\{m_i\}$.\par Here the violation of quasi-locality and thus the non Gibbsiannes of $\mu^B(\{m_i\})$ is due to the presence, for large enough $\beta$ and arbitrary value of $h$, of a first order phase transition in the multicanonical model represented by the object system constrained to have zero magnetization in each block $Q_i$. Notice that, since the local magnetizations are fixed, the value of the magnetic field is irrelevant. Although the proof of this result was given only for the case of 2x2 blocks, it seems quite plausible that it persists for any value of the side of the blocks $Q_i$, thus excluding the possibility of restoring Gibbsiannes by simply enlarging the side of the blocks, in contrast to what happens for the decimation trasformation.\par On the other hand, if, for example, the magnetic field is large, the object system without constraints is very close to a product measure and the non Gibbsian measure $\mu^B(\{m_i\})$ itself enjoys nice mixing properties like exponential decay of truncated correlations functions and, quite likely, a weaker version of quasi-locality, as the one introduced in [FP] . Moreover it is quite obvious that the event of having zero magnetization in each block is exceptional and thus, in some sense, the above pathology should be "unstable" with respect to a little bit of decimation. This kind of considerations was already suggested, on a informal level, in [EFS] (see page 1066).\par In the present note we pursue the above point of view quite seriously, since, in our opinion, the "stability" or "instability" of non-Gibbsiannes of a measure under decimation is a relevant property. Decimation in fact corresponds to select certain variables, which are the only "relevant" ones for the kind of questions one is interested in, and disregard (integrate out) the "irrelevant" ones; moreover important thermodynamic quantities like the free energy (and their analyticity properties) or the asymptotic behaviour of the truncated correlations can be computed equally well with the decimated measure. Thus, if Gibbsiannes can be restored with the help of some decimation, then the pathologies decribed above becomes irrelevant at least as far as certain variables are concerned . On the contrary, if the measure $\mu$ under cosideration is non-Gibbsian and remains of such a type after an arbitrary decimation, then such a character becomes, in our opinion, a much more important feature of the system described by $\mu$, probably related to some non trivial long range dependence hidden inside the system itself. \par In this note we illustrate in full details the above considerations, first for the block spin Ising model (see section 2) and then for the invariant measure of a certain stochastic dynamics on the configuration space $\{0,1\}^{\bf Z^2}$ (see section 3). In the first case we prove that, if we decimate the block spin model on the even blocks (see section 2 for details) and we take the external field $h$ large enough, then we end up with a nice, weakly coupled, Gibbs measure whose potential is expressed via a convergent cluster expansion. To perform the calculation we use the commutativity of the decimation with the block spin trasformation, that is we first decimate and then apply the block spin transformation. \par In the second case we show that, independently of the side of the blocks of the decimation, the decimated measure remains non Gibbsian like the starting measure.\par After the present paper was completed, we learnt of a recent work by A. v. Enter, R. Fern\'andez and R. Koteck\'y where, in particular, the authors establish non-Gibbsianness of the renormalized measure $\n \; = \; T_b ^{m r} \m _{\b, h}$ obtained by applying the majority rule transformation, over blocks of side $b$, to the Ising Gibbs measure $\m _{\b, h}$ for $\b$ and $h$ large enough.\par One can easily check that, by the same methods developed in Section 2 of the present paper , it is possible to restore Gibbsianness by simply decimating the measure $\n$ over the even blocks ( see Section 2). In other words a statement analogous to the one of Theorem 2.1 holds true.\par \bigskip {\bf Acknowledgements.}\par We are very grateful to Aernout van Enter for informing us about their work prior to publication. \pagina \numsec=2\numfor=1 {\bf Section 2. The block spin and decimation transformation.} \vskip 1cm In this main section we discuss in details the effect of a decimation over the odd blocks (see below) on the block spin Ising model for which {\it non Gibbsiannes} was proved in [EFS]. For simplicity we restrict ourselves to the case of large external magnetic field (but see the remark at the end of the section for more general situations).\par We show that, after the decimation, Gibbsiannes is recovered. As already explained in the introduction, a key remark is that the two transformations, the block spin and the decimation, commute so that we can first decimate the original Gibbs measure of the Ising model and then do the block average transformation. The technical tool is the cluster expansion that provides naturally all the necessary cancellations. Let us start with the details. \par The Ising hamiltonian in a volume $\L \in {\bf Z^d}$ with open (empty) or periodic boundary conditions is given by : $$ H_{\L} (\s_{\L}) \; = \; - J/2 \;\sum _{x,y \in \L} \;\s_x \s_y \;\; - h/2\; \sum _{x \in \L} \; \s_x \Eq(1.1) $$ where $ \s_{\L} \; \in \; \O_{\L} \; \equiv \; \{ -1, +1 \}^{\L}$ and $h$ is the external magnetic field. \par The corresponding finite volume Gibbs measure at inverse temperature $\b$ and magnetic field $h$ is given by : $$ \m_{\L} \; = \;{\exp[\; -\b ( H_{\L} (\s_{\L}) ) \;]\over Z_{\L} } \Eq(1.1a) $$ where the normalization factor $$ Z_{\L} \; = \; \sum _{\s_{\L} \in \O_{\L} } \exp[\; -\b ( H_{\L} (\s_{\L}) ) \;] \Eq(1.1b) $$ is called {\it partition function}. \par We will consider the case when both the inverse temperature $\b$ and the external magnetic field $h$ are very large.\par For the sake of simplicity of the exposition we will assume that the dimension is $d=2$.\par Consider the partition of ${\bf Z^2}$ into $2\times 2$ squared blocks $Q_i$ of side 2 (each containing 4 sites). A block $Q_i$ can be characterized by its leftmost down site $x(Q_i)$. The $x(Q_i)$ are of the form : $$ x(Q_i) \equiv x^{(i)} \equiv ( x^{(i)}_1,x^{(i)}_2 )\; ; \; x^{(i)}_1 = 2 y^{(i)}_1,x^{(i)}_2 = 2 y^{(i)}_2 \;\hbox{with}\;\; y^{(i)} \; \equiv \; (y^{(i)}_1,y^{(i)}_2) \in {\bf Z^2} \Eq(1.1c) $$ We write $$ Q_i = Q(y^{(i)}) \;\; \hbox{if} \;\;y^{(i)} = x(Q_i) /2 \Eq(1.2) $$ Now we introduce a partition the lattice ${\bf Z^2}$ into two sublattices ${\bf Z^2_e}$ and ${\bf Z^2_o}$ ( the subscripts $e,o$, respectively, stand for even and odd). They are given by : $$ {\bf Z^2_{e,o}} = \{ y \equiv (y_1,y_2) \in {\bf Z^2} \; : \; y_1+ y_2 = \hbox{even integer, odd integer} \} \Eq(1.3) $$ Given a $2\times 2$ block $Q_i$ we call it even or odd according to the sublattice to which $ y^{(i)} = x(Q_i) /2$ belongs.\par We decompose the original lattice ${\bf Z^2}$ into the union $$ {\bf Z^2} = {\cal A} \cup {\cal B} \Eq(1.4) $$ where ${\cal A} = \cup _i A_i$ is the set of the even blocks $A_i \;\; \hbox{with}\;\; x(A_i) /2\; \in \;{\bf Z^2_e}$ and ${\cal B} = \cup _i B_i$ is the set of the odd blocks $B_i \;\; \hbox{with}\;\; x(B_i) /2 \in {\bf Z^2_o}$. Notice that in our notation we suppose that the total set $ {\cal Q} =\cup _i Q_i$ of the $2\times 2$ blocks as well as the sets ${\cal A} = \cup _i A_i$ , ${\cal B} = \cup _i B_i$ of even and odd, respectively, blocks is given a certain order, for example the lexicographic one, but this ordering will never be used explicitely.\par We use the notation $\a_i,\; \b_i$ to denote, respectively, the spin configurations inside the blocks $ A_i, B_i $; $\a_i ,\; \b_i$ take $ 2^4 = 16$ possible values.\par We denote by $e_1,e_2,e_3,e_4$ the unit vectors $ (0,1), (1,0), (-1,0), (0,-1)$\par Given a bkock $B_i \equiv Q_{l(i)} \hbox {with} \; x(Q_{l(i)}) /2 \; =\;y^{(l(i))}$, we denote by $ A^1_i, A^2_i, A^3_i, A^4_i$ the four nearest neighbours $A$-blocks given by $ A^j_i = Q(y^{l((i))}+ e_j); \; j=1,\dots , 4\;$ and by $\a_i^1,\a_i^2, \a_i^3, \a_i^4$ the corresponding spin configurations. We use $ \underline \a_i$ to denote the set of spin configurations $\a_i^1,\a_i^2, \a_i^3, \a_i^4$ in these four $A$-blocks. \par By $Z_{B_i}^{ \underline \a^i}$ we denote the partition function in the block $B_i$ with boundary conditions given by $ \underline \a_i$ .\par We call $0$ a fixed reference configuration inside a $2 \times 2 $ block; for instance $0$ can be chosen to be the configuration with all minus spins inside the block. We use $\underline 0$ to denote the configuration $\underline \a _i \;\equiv \a^j_i = 0 \; \forall \; j =1,\dots, 4$ in the set $ A^1_i, A^2_i, A^3_i, A^4_i$ of nearest neighbours $A$-blocks to a given $B$-block $B_i$.\par Given an integer $L$ multiple of $4$, consider the squared box $\L \; \equiv \L_L \; \equiv \; [-L/2, L/2 +1]^2$.\par We choose, for simplicity, periodic boundary conditions; namely $\L$ , by identifying its opposite sides, becomes a two-dimensional finite torus. Any other boundary condition could be considered as well, with only minor changes.\par Using simply $\a,\;\b$ to denote the global spin configuration in all the $A$-blocks , $B$-blocks, respectively, contained in $\L$, we can write the following expression for the partition function in $\L$ ( with periodic boundary conditions) : $$ Z_{\L} \;= \; \sum _{\a} Z_{\L} ( \a) \Eq(1.4a) $$ where: $$ Z_{\L} ( \a)\;=\;\exp ( \sum_{A_i\subset \L} H(\a_i) ) \prod_{B_i \subset \L} Z_{B_i}^{\underline \a _i}\Eq(1.4b) $$ and, for $ \a_i \; = \;\s_{A_i}$, $H(\a_i) $ is the self-energy inside the block $A_i$: $$ H(\a_i) \; \equiv \;H_{A_i}(\s_{A_i})\; = \; - J/2 \;\sum _{x,y \in A_i} \;\s_x \s_y \;\; - h/2\; \sum _{x \in A_i} \; \s_x \Eq(1.4c) $$ \equ (1.4a) is simply obtained by a decimation procedure; namely by first summing over the $\b$-variables keeping fixed the $\a$-variables which play the role of fixed boundary conditions . The sum over the $\b$-variables, for fixed $\a$, is immediately seen to factorize into independent sums over the single $\b_i$ which are mutually decoupled because of the form (nearest neighbour) of the Ising interaction.\par By simple manipulations of the previous expression we get: $$ Z_{\L}\; = \; \prod_{B_i\subset \L} ({1 \over Z_{B_i}^{\underline 0} })^3 \sum _{\a} \exp (\sum_i H(\a_{A_i\subset \L} ) ) $$ $$\prod _{B_i\subset \L} ({ Z_{B_i}^{\a_i^1,\a_i^2, \a_i^3, \a_i^4} Z_{B_i}^{ \underline 0} Z_{B_i}^{ \underline 0} Z_{B_i}^{ \underline 0} \over Z_{B_i}^{\a_i^1,0, 0, 0} Z_{B_i}^{0,\a_i^2, 0, 0} Z_{B_i}^{0,0, \a_i^3,0} Z_{B_i}^{0,0,0, \a_i^4}} -1 +1) Z_{B_i}^{\a_i^1,0, 0, 0} Z_{B_i}^{0,\a_i^2, 0, 0} Z_{B_i}^{0,0, \a_i^3,0} Z_{B_i}^{0,0,0, \a_i^4} \Eq(1.5) $$ and we define $$ \widetilde Z_{\L} \; = \; \prod _{B_i\subset \L } ({1 \over Z_{B_i}^{\underline 0} })^3 \prod _{A_i\subset \L} \sum _{\a_i} ( \exp(H(\a_i) Z_{B_i^3}^{\a_i^1,0, 0, 0} Z_{B_i^4}^{0,\a_i^2, 0, 0} Z_{B_i^2}^{0,0,0, \a_i^3} Z_{B_i^1}^{0,0, \a_i^4, 0} ) \Eq(1.6) $$ where if $ A_i \equiv Q_{m(i)}$, we set $B^j_i = Q(y^{m((i))}+ e_j)' \; j=1, \dots , 4 \;$ and we use the short forms $\sum_{\a_i}, \sum_{\a}$, to denote $\sum_{\a_i\in \O_{A_i}}, \sum_{\a \in \O_{\cal A}}$, respectively. \par A useful graphical way to describe the r.h.s. of \equ (1.6) is to associate to anyone of the four partition functions appearing in the r.h.s. of the \equ (1.5), namely to $Z_{B_i}^{\a_i^1,0, 0, 0}$, $Z_{B_i}^{0,\a_i^2, 0, 0}$, $Z_{B_i}^{0,0, \a_i^3,0}$, $Z_{B_i}^{0,0,0, \a_i^4}$, four arrows, emerging from the block $B_i$ and ending in the blocks $ A^1_i, A^2_i, A^3_i, A^4_i$ ; namely four arrows parallel to the four unit vectors $e_1,e_2,e_3,e_4$, respectively. Then in \equ (1.6) appear the terms (partition functions) corresponding to the four arrows ending into the $A$-block $A_i$ and emerging from the four nearest neighbour $B$-blocks.\par Consider now, for every $A$-block $A_i$, the probability measure on $\a_i$ given by: $$ \n (\a_i) \; =\; { \exp (H(\a_i)) Z_{B_i^3}^{\a_i^1,0, 0, 0} Z_{B_i^4}^{0,\a_i^2, 0, 0} Z_{B_i^2}^{0,0, \a_i^3,0} Z_{B_i^1}^{0,0,0, \a_i^4} ) \over \sum_{\a_i} \exp (H(\a_i)) Z_{B_i^3}^{\a_i^1,0, 0, 0} Z_{B_i^4}^{0,\a_i^2,0, 0} Z_{B_i^2}^{0,0, \a_i^3,0} Z_{B_i^1}^{0,0,0, \a_i^4} ) } \Eq (1.7) $$ We set $$ \hat Z _{A_i}\; =\; \sum_{\a_i} \exp (H(\a_i)) Z_{B_i^3}^{\a_i^1,0, 0, 0} Z_{B_i^4}^{0,\a_i^2, 0, 0} Z_{B_i^2}^{0,0, \a_i^3,0} Z_{B_i^1}^{0,0,0, \a_i^4} ) \; \equiv \; Z^{\underline0}_{V_i} \Eq(1.8) $$ where $$V_i = A_i \cup B_i^1 \cup B_i^2 \cup B_i^3 \cup B_i^4 \Eq(1.9) $$ >From \equ (1.6), \equ (1.8) we obtain: $$ \widetilde Z \;= \; \prod _{B_i \subset \L} ({1 \over Z_{B_i}^{\underline 0} })^3 \prod _{A_i\subset \L} \hat Z _{A_i} \Eq(1.10) $$ from \equ (1.5) ,\equ (1.7), \equ (1.8), \equ (1.10), we get: $$ Z_{\L}\; = \; \widetilde Z _{\L} \sum _{\a} \prod _{A_j \subset \L} \n(\a_j) \prod _{B_i \subset \L}( \psi _{B_i} ( \a_i^1, \a_i^2, \a_i^3, \a_i^4) + 1) \Eq(1.11) $$ where $$\psi _{B_i} ( \a_i^1, \a_i^2, \a_i^3, \a_i^4) \equiv { Z_{B_i}^{\a_i^1,\a_i^2, \a_i^3, \a_i^4} Z_{B_i}^{ \underline 0} Z_{B_i}^{ \underline 0} Z_{B_i}^{ \underline 0} \over Z_{B_i}^{\a_i^1,0, 0, 0} Z_{B_i}^{0,\a_i^2, 0, 0} Z_{B_i}^{0,0, \a_i^3,0} Z_{B_i}^{0,0,0, \a_i^4}} -1 \Eq(1.12) $$ We define the {\it renormalized hamiltonian} as $$ H^r_{\L}\; = \; \sum _{A_i\subset \L} H(\a_i) + \sum _{B_i \subset \L} - \;\log Z_{B_i}^{\a_i^1,\a_i^2, \a_i^3, \a_i^4} \Eq(1.13) $$ Then after having extracted the one-body part we get: $$ H^r_{\L}\; = \; \sum _{A_i \subset \L} H(\a_i) + \sum _{B_i \subset \L} -\;(\log Z_{B_i^3}^{\a_i^1,0, 0, 0}+ \log Z_{B_i^4}^{0,\a_i^2, 0, 0}+ \log Z_{B_i^2}^{0,0, \a_i^3,0} + \log Z_{B_i^1}^{0,0,0, \a_i^4}) \; + $$ $$ +\sum_{B_i \subset \L}- \; \log ( { Z_{B_i}^{\a_i^1,\a_i^2, \a_i^3, \a_i^4} Z_{B_i}^{ \underline 0} Z_{B_i}^{ \underline 0} Z_{B_i}^{ \underline 0} \over Z_{B_i}^{\a_i^1,0, 0, 0} Z_{B_i}^{0,\a_i^2, 0, 0} Z_{B_i}^{0,0, \a_i^3,0} Z_{B_i}^{0,0,0, \a_i^4} }) + \hbox { const.} \Eq(1.14) $$ Now we want to use the expression given in \equ (1.11) to make the second step; namely the sum over the spin configuration $\a_i \; \equiv \; \s_{A_i}$ with given values $m_i$ of the magnetizations $ m_{A_i}\; \equiv \; m_{A_i} (\a_i)\;\; \equiv \; \sum _{x \subset A_i} \s _x$ in the blocks $A_i$.\par If $N =N(L)$ is the total number of $A$-blocks in $\L$, let $$ Z_{\L} ( m_1, \dots , m_N)\; = \; \sum_{\a}\prod _{A_i \subset \L} ({\bf 1}_{ m_{A_i} = m_i}(\a_i) ) Z_{\L}(\a) \Eq(1.15) $$ where $$ {\bf 1}_{ m_{A_i} = m_i}(\a_i) = 1 \;\;\hbox {if }\;\; m_{A_i}\; \equiv m_{A_i}(\a_i) = m_i;\;\;\;\;\; {\bf 1}_{ m_{A_i} = m_i}(\a_i)\; \;=\;0 \;\;\hbox {otherwise} \Eq(1.15a) $$and $Z_{\L}(\a)$ has been defined in \equ (1.4b) ).\par Now we transform our original block spin system, described by \equ (1.15), into a polymer system; for this purpose we need some definitions.\par A ( four-body) {\it bond } $p_i \; \equiv \; p(B_i)$, for a given $B_i$, is the set $ p_i\; = \; \{A^1_i, A^2_i, A^3_i, A^4_i\}$ of the four $A$-blocks nearest neighbours to $B_i$.\par Its {\it support} $\tilde p_i$ is given by $ \tilde p_i\; = \; \cup_{j=1} ^4 A_i^j$ \par A set of bonds $R\; = \; p_i, \dots , p_k$ is called {\it polymer} if it is {\it connected} in the sense that for every pair $ p_i, p_j \; \in \; R$ there exists a chain $ p_{k_1}, \dots , p_{k_l} \; \in \; R; \;$ with $p_{k_1} \; = p_i, \; \; p_{k_l} \; = \; p_j $ of bonds which are connected in the sense that $ \tilde p_{k_i}\cap \tilde p_{k_{i+1}} \neq \emptyset,\; i=1, \dots , l$.\par We call {\it support} of a polymer $R$ and we denote it by $\widetilde R $ the union of the supports $ \tilde p_i$ of the bonds $ p_i \; \in \; R$. \par We say that two polymers $R_1$ and $R_2$ are {\it compatible} if $\widetilde R_1 \cap \widetilde R_2 \; = \; \emptyset $.\par We set $$ \widetilde \z_{R} (\a_{R} )\; =\; \prod _{p(B_i)\in R} \psi _{B_i} ( \a_i^1, \a_i^2, \a_i^3, \a_i^4) \Eq(1.15b) $$ Now let : $$ \bar Z_{\L}(m_1, \dots , m_N) \; =\; \prod _{A_i \subset \L} \bar Z_{A_i} (m_i) \Eq(1.16) $$ with $$ \bar Z_{A_i} (m_i)\; = \; \sum _{\a_i} \n(\a_i) {\bf 1}_{ m_{A_i} = m_i}(\a_i) \Eq(1.17). $$ Let $$ \m_{m_i} (\a_i) \; \equiv \; 1/ \bar Z_{A_i} (m_i) \n (\a_i) {\bf 1}_{ m_{A_i} = m_i}(\a_i). \Eq(1.18) $$ >From \equ (1.4a),\equ (1.4b),\equ (1.5), \equ (1.11),\equ (1.15),\equ (1.16),\equ (1.17) it is easy to get: $${ Z_{\L} ( m_1, \dots , m_N)\; \over \widetilde Z_{\L} \bar Z_{\L}(m_1, \dots , m_N) }\; = \; \prod _{A_i \subset \L} \m_{m_i} (\a_i) ( 1 + \sum _{n \geq 1} \sum _{R_1, \dots ,R_n: \atop \widetilde R_i \subseteq \L, \widetilde R_i \cap \widetilde R_j = \emptyset} \prod_{i=1}^n \widetilde \z_{R_i} (\a_{R_i} ) )\Eq(1.19) $$ Now, if we introduce the {\it activity} $\z (R)\,\equiv\,\z_m(R)$ of a generic polymer $R$ as : $$ \z (R)\; = \; \prod _{A_i \in \widetilde R} \sum _{ \a_i } \m_{m_i} (\a_i) \widetilde \z_{R} (\a_{R} ) ,\Eq(1.20) $$ we can write: $$ { Z_{\L} ( m_1, \dots , m_N)\; \over \widetilde Z_{\L} \bar Z_{\L}(m_1, \dots , m_N) }\; =\; 1 + \sum _{n \geq 1} \sum _{R_1, \dots ,R_n: \atop \widetilde R_i \subseteq \L, \widetilde R_i \cap \widetilde R_j = \emptyset} \prod_{i=1}^n \z (R_i) \Eq(1.21) $$ Now we observe that in the region of thermodynamic parameters that we are considering, namely $h$ and $\b$ large, the activity of our polymers is indeed very small, uniformly in the $m_i$'s, in the proper sense so that we can apply the theory of the cluster expansion and obtain its convergence. As we will discuss later on, one could simply assume $ h \neq 0$ and $\b$ sufficiently large provided the $2 \times 2$ blocks are replaced by blocks of large enough side ( depending on $h$and $\b$). \par In our case of $h$ very large everything is much simpler since after decimation our system is weakly coupled on scale one.\par In particular it is easy to get the following statement:\par\bigskip {\bf Proposition 2.1 .} \par \bigskip For every $\e >0$ there exists a value $h(\e)$ of the magnetic field such that if $ h \; > \; h(\e)$ and $\b$ is sufficiently large, $$ \sup _{ \a_i^1, \a_i^2, \a_i^3, \a_i^4} |\psi _{B_i} ( \a_i^1, \a_i^2, \a_i^3,\a_i^4) |\; < \; \e \Eq(1.22) $$ \bigskip >From the previous proposition one can easily deduce, by standard methods, the convergence of a cluster expansion of the thermodynamic as well as the correlation functions . The elemetary geometric objects of this expansion will be clusters of incompatible polymers (see, for instance, [GMM] for more details). Many properties for the image block-spin system ( after decimation over the odd $B$-blocks), described by the probability measure with weights proportional to $Z_{\L} ( m_1, \dots , m_N)$, can be deduced using this convergent cluster expansion.\par We will concentrate now on one important feature of this measure namely its {\it Gibbsiannnes } .\par First of all let us introduce the doubly renormalized hamiltonian, after decimation on the odd $B$-blocks and block-averaging over the surviving even $A$-blocks. We will denote it by $ H^{b,d}_{\L} ( m_1, \dots , m_N)$ where in the superscript $b,d $ stand for block spin averaging transformations and for decimation transformation, respectively.\par We define it as $$ H^{b,d}_{\L} ( m_1, \dots , m_N) \; = \; - \;\log ({ Z_{\L} ( m_1, \dots , m_N)\; \over \widetilde Z_{\L} \bar Z_{\L}(m_1, \dots , m_N) }) \Eq(1.23) $$ which corresponds to a particular choice of the zero of the energy for our doubly renormalized system (namely obtained by decimation and block spin transformsation).\par Our doubly renormalized probability measure is : $$ \m_{\L} ^{b,d} ( m_1, \dots , m_N) ={ Z_{\L} ( m_1, \dots , m_N) \over \sum_{m_1, \dots , m_N} Z_{\L} ( m_1, \dots , m_N)} $$ $$ \equiv {\exp (- H^{b,d}_{\L} ( m_1, \dots , m_N) ) \over \sum_{m_1, \dots , m_N} \exp (- H^{b,d}_{\L} ( m_1, \dots , m_N) )} \Eq(1.23a)$$ We want now to state and prove our main result about the gibbsianness of our doubly renormalized measure $ \m ^{b,d}$ in the thermodynamic limit $\L \to {\bf Z^2}$ . \par\bigskip {\bf Theorem 2.1 .} \par\bigskip If the external magnetic field $h$ and the inverse temperature $\b$ are sufficiently large: \par\noindent i) There exists the (weak) thermodynamic limit limit of the finite volume doubly renormalized measure : $$ \m ^{b,d} ( m_1, \dots , m_N)\;\; = \; \lim _ {\L \to {\bf Z^2}} \m_{\L} ^{b,d} ( m_1, \dots , m_N)\; \Eq(1.23b) $$ \noindent ii) $ \m ^{b,d} ( m_1, \dots , m_N)$ is a Gibbs measure corresponding to a finite norm, translationally invariant, potential.\par \bigskip {\it Proof.}\par \bigskip The proof of the Theorem is based on the theory of cluster expansion applied to the system of polymers described by \equ (1.21) For this purpose let us recall a proposition which summarizes the basic results on cluster expansion that we need to prove Gibbsianness.\par The proof of the proposition together with more details can be found in [CO]. \par \bigskip {\bf Proposition 2.2} \par \bigskip Let $\;\Xi_\L\;$ denote the polymer partition function: $${\Xi}_{\L}=1+\sum_{k\geq 1}\;\sum_{{R_1,\dots ,R_k:\atop{\widetilde R}_i\subseteq\L,1\leq i\leq k,}\atop{\widetilde R}_i\cap{\widetilde R}_{i'} =\emptyset ,1\leq i\, 4$, e.g. $C\,=\,5$. With this choice one has in fact that the Hamiltonian of a single block $B_i$ has a unique ground state identically equal to plus one, independently of the boundary conditions $\a_i^1\dots \a_i^4$. In particular it follows that $$\lim_{\beta \to \infty} \psi (\a_i^1\dots \a_i^4)\;=\;0$$ and the convergence of the cluster expansion can again be proved. \par Of course, in the above argument the fact that the unique ground state is the special configuration identically equal to plus one is completely irrelevant. Thus the above method is able to treat systems at low enough temperature having the property that the ground state in a large enough volume is unique and independent of the boundary conditions. \par One may also want to consider much more general cases in which the thermodynamic parameters garantee only a strong form of weak dependence of the finite volume Gibbs measure on the boundary conditions, that we call {\it Strong Mixing} (see e.g. [MOS] and references therein). This is the case for example of 2D ferromagnetic systems just above the critical point [MOS]. In such more general situations, even if the side of the blocks is large, the decimation over the odd blocks may be not enough to depress some long range dependence in the doubly renormalized measure and one is forced to decimated further. One may stop the extra decimation until each surviving blocks, namely the blocks of the final block spin transformation, are separated one from the other by at least one block of the decimation. We shall omit the details of these computations that are however quite similar to the ones exposed above. \vskip.5cm \numsec=3\numfor=1 {\bf Section 3. An example of persistence of non-Gibbsianness under decimation.} \vskip.5cm \par \vskip.5cm In this last section we briefly discuss another example of a measure $\mu$ on $\{0,1\}^{{\bf Z}^2}$ with nicely decaying correlations functions, which is {\it non Gibbsian} and remains such even after decimation on blocks of arbitrary side. The example comes from a model of random discrete time dynamics introduced in [MS] and further analyzed in [M].\par The setting is as follows: to each point x in the lattice $ {\bf Z}^2$ we associate an occupation variable $\sigma (x)$ with values 0 or 1; given a configuration $\sigma \; \in\; \{0,1\}^{{\bf Z}^2}$ we then define its clusters as the maximal connected sets of sites in which the configuration $\sigma $ is equal to one, where a set $C\,\subset \, {\bf Z}^2$ is connected if for any pair of sites $x,y\;\in \;C$ there exists a path $x_1,x_2\dots x_n$ of sites in $C$ such that $$x_1\,=\,x,\;x_n\,=\,y,\; \hbox{and } \vert x_i-x_{i+1}\vert\,=\,1\; i=1\dots n-1$$ With this position the dynamics goes as follows: given the configuration $\sigma _t\;\in\; \{0,1\}^{{\bf Z}^2}$ at time t, in order to define the new configuration $\sigma _{t+1}$ at time t+1, we first remove each cluster of $\sigma _{t}$ independently one from the other with probability 1/2 ; as a second step we create particles in each empty site independently with probability p. \par For shortness we will refer to the first part of the updating as the annihilation of particles and to the second part as the creation of particles. Note that both processes occur simultaneously (i.e. the updating is parallel) and that the non trivial interaction of the model is all contained in the killing process .\par The above dynamics is similar to a model considered by Graannan and Swindle in [GS] although in their model clusters disappear with a rate proportional to their size. The two main results that we need from [MS] and [M] are the following:\bigskip {\bf Theorem 3.1 .}\par \item{i)} For p sufficiently small there exists a unique invariant measure $\mu$ on $\{0,1\}^{{\bf Z}^2}$ for the above dynamics and its truncated correlations decay faster than any inverse power (see corollary 5.1 in [M]) \item{ii)} If $\Omega_N$ denotes the event that the cube $\L_N$ of side N centered at the origin is filled with particles, then there exists a constant $c$ such that: $$\mu (\Omega_N)\; \geq \; \exp(-cN) $$ (see theorem 5.1 in [MS]). \bigskip In particular part ii) of the above theorem implies that the measure $\mu$ cannot be the Gibbs measure for any absolutely summable interaction, since it has {\it wrong} large deviations: an exponential of the surface instead of an exponential of the volume (see e.g. [EFS] for more details). \bigskip {\bf Remark} It is important to observe that the measure $\mu$ is non Gibbsian because of reasons which are very different from the ones behind the {\it non Gibbsiannes} of the block spin (without decimation) Ising model discussed in section 2. There Gibbsiannes is lost due to the fact that, conditioned to the event of having zero magnetization in each 2x2 block, the original spin model undergoes a phase transition (see [EFS]). Here, on the contrary, Gibbsiannes is violated because the measure $\mu$ has zero relative entropy density with respect to the $\d$-measure concentrated on the configuration identically equal to plus one (see [EFS]). Clearly the latter is non Gibbsian because it violates the nonnullity condition or absence of hard-core interaction (see [EFS] 4.5.5 and 2.3.3). We actually suspect that $\mu$ itself violates the nonullity condition but we do not have a formal proof of this fact.\par Keeping in mind such a difference between the block spin Ising model and the present one, it is not entirely surprising that a decimation can suppress the phase transition in the first one and thus restore the Gibbsiannes, while it is useless in this new situation where the non Gibbsiannes is due to a much stronger and rigid phenomenon.\bigskip We want now to apply the usual decimation described in section 2 to the measure $\mu$ and show that it leads to a new measure $\mu^{d,b_l}$ which is again non Gibbsian irrespectively of the side $l$ of the blocks on which the decimation acts. \par As before let us consider the partition of ${\bf Z^2}$ into $l\times l$ squared blocks $Q_i$ of integer side $l$ and let us decompose the lattice ${\bf Z^2}$ into the two odd and even sublattices ${\bf Z^2_e}$ and ${\bf Z^2_o}$ namely $$ {\bf Z^2} = {\cal A} \cup {\cal B} \Eq(3.1) $$ where ${\cal A} = \cup _i A_i$ is the set of the even blocks and ${\cal B} = \cup _i B_i$ is the set of the odd blocks (see section 2 for more details). Let us also call $\mu^{d,b_l}$ the projection or relativization of the measure $\mu$ to ${\cal A}$. Then we have : \bigskip {\bf Theorem 3.2 .} \par For any value of the decimation parameter $l$ the measure $\mu^{d,b_l}$ is not Gibbsian for any absolutely summable interaction.\bigskip {\bf Proof} \par Let us denote with $\Omega_N^A$ the event that all the even blocks $A$ contained inside the cube $\L_N$ of side N centered at the origin are filled with particles. Then we trivially have: $$\mu^{d,b_l} (\Omega_N^A)\;=\;\mu (\Omega_N^A)\;\geq\;\mu (\Omega_N)\; \geq \; exp(-cN) $$ Thus also the decimated measure $\mu^{d,b_l}$ has wrong large deviations and the result follows.\par \bigskip {\bf Remark} \par \bigskip We want to remark that, by the same argument used before for the decimation in one of the two L-block sublattices, it is immediate to prove that the non-Gibbsianness of our measure $\m$ persists under any {\it extensive} decimation; namely a decimation such that the surviving spins, even though they can be very sparse, have a well defined volume density. \vskip 1cm \centerline{\bf References}\bigskip\noindent \refj{BMO}{G.Benfatto, E.Marinari, E.Olivieri} {Some numerical results on the block spin transformation for the 2D Ising model at the critical point}{CARR preprint}{}{}{1994} \refj{CO}{M. Cassandro, E. Olivieri} {Renormalization group and analyticity in one dimension: a proof of Dobrushin's theorem }{Commun. Math. 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