BODY %VERSION WITH SOME FIGURES MISSING %%%%%%%%%Definizione dei caratteri % Definisco i caratteri BIG in ROMAN \font\biga=CMR10 scaled \magstep1 \font\bigb=CMR10 scaled \magstep2 \font\bigc=CMR10 scaled \magstep3 \font\bigd=CMR10 scaled \magstep4 \font\bige=CMR10 scaled \magstep5 % Definisco i caratteri BIG in GRASSETTO \font\bigbfa=CMBX10 scaled\magstep1 \font\bigbfb=CMBX10 scaled\magstep2 \font\bigbfc=CMBX10 scaled\magstep3 \font\bigbfd=CMBX10 scaled\magstep4 \font\bigbfe=CMBX10 scaled\magstep5 % Definisco i caratteri SMALL in ROMAN \font\sma=CMR9 \font\smb=CMR8 \font\smc=CMR7 \font\smd=CMR6 \font\sme=CMR5 % Definisco i caratteri SMALL in GRASSETTO \font\smbfa=CMBX9 \font\smbfb=CMBX8 \font\smbfc=CMBX7 \font\smbfd=CMBX6 \font\smbfe=CMBX5 %%%%%%%%%%%%%%%%%%% Tutte le macros che uso nella tesi %%%%%%%%%%%%%%%%%%% \def\pa{pathwise approach} \def\ret{\Lambda_{N}} \def\volret{\vert\ret\vert} \def\eps{\varepsilon} \def\menouno{-{\underline 1}} \def\piuuno{+{\underline 1}} \def\zero{{\underline 0}} \def\aaa{{\cal A}} \def\faaa{\partial{\cal A}} \def\prot{{\cal P}} \def\bc{Blume--Capel} \def\grle{{\scriptscriptstyle {>\atop <}}} \def\fb{\partial B} \def\fd{\partial {\cal D}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%% Rettangoli %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %rect 'rettangolo' \def\rect#1#2{{\vcenter{\vbox{\hrule height.3pt \hbox{\vrule width.3pt height#2truecm \kern#1truecm \vrule width.3pt} \hrule height.3pt}}}} %birect 'doppio rettangolo' %#1 #2 'dimensioni di quello esterno' %#3 #4 'dimensioni di quello interno' %#5 #6 'margine sinistro e destro' %#7 'di quanto viene sollevato il rettangolo interno' \def\birect#1#2#3#4#5#6#7{{\vcenter{\vbox{\hrule height.6pt \hbox{\vrule width.6pt height#2truecm \kern#5truecm {\raise#7truecm\vbox{\hrule height.6pt \hbox{\vrule width.6pt height#4truecm \kern#3truecm \vrule width.6pt} \hrule height.6pt}} \kern#6truecm\vrule width.6pt} \hrule height.6pt}}}} %% Per disegnare le cornici e' conveniente usare i seguenti valori %% dei parametri: %% %% #3 = #1 - 1 %% #4 = #2 - 1 %% #5 = #6 = #7 = 0.5 \def\square{\rect{0.2}{0.2}} %%%%%%%%%%%%%%% FORMATO \magnification=\magstep1\hoffset=0.cm \voffset=1truecm\hsize=16.5truecm \vsize=21.truecm \baselineskip=14pt plus0.1pt minus0.1pt \parindent=12pt \lineskip=4pt\lineskiplimit=0.1pt \parskip=0.1pt plus1pt \def\ds{\displaystyle}\def\st{\scriptstyle}\def\sst{\scriptscriptstyle} \font\seven=cmr7 %%%%%%%%%%%%%%%% GRECO \let\a=\alpha \let\b=\beta \let\c=\chi \let\d=\delta \let\e=\varepsilon \let\f=\varphi \let\g=\gamma \let\h=\eta \let\k=\kappa \let\l=\lambda \let\m=\mu \let\n=\nu \let\o=\omega \let\p=\pi \let\ph=\varphi \let\r=\rho \let\s=\sigma \let\t=\tau \let\th=\vartheta \let\y=\upsilon \let\x=\xi \let\z=\zeta \let\D=\Delta \let\F=\Phi \let\G=\Gamma \let\L=\Lambda \let\Th=\Theta \let\O=\Omega \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)}} \tolerance=10000 \centerline {\bf METASTABILITY AND NUCLEATION} \centerline {\bf FOR THE BLUME-CAPEL MODEL.} \centerline {\bf DIFFERENT MECHANISMS OF TRANSITION.} \vskip 2 truecm \centerline {Emilio N. M. Cirillo}\par\noindent \vskip 0.5 truecm \centerline {\it Dipartimento di Fisica dell'Universit\`a di Bari and}\par\noindent \centerline {\it Istituto Nazionale di Fisica Nucleare, Sezione di Bari.}\par\noindent \centerline {\it V. Amendola 173, I-70126 Bari, Italy.}\par\noindent \centerline {\rm E-mail: cirillo@axpba0.ba.infn.it} \vskip 1 truecm \centerline {Enzo Olivieri}\par\noindent \vskip 0.5 truecm \centerline {\it Dipartimento di Matematica - II Universit\`a di Roma}\par\noindent \centerline {\it Tor Vergata - Via della Ricerca Scientifica - 00173 ROMA - Italy}\par\noindent \centerline {\rm E-mail: olivieri@mat.utovrm.it} \vskip 1.5 truecm \centerline {\bf Abstract} \vskip 0.5 truecm \centerline { \vbox { \hsize=13truecm \baselineskip 0.35cm We study metastability and nucleation for the Blume-Capel model: a ferromagnetic nearest neighbour two-dimensional lattice system with spin variables taking values in $\{-1,0,+1\}$. We consider large but finite volume, small fixed magnetic field $h$ and chemical potential $\lambda$ in the limit of zero temperature; we analyze the first excursion from the metastable $-1$ configuration to the stable $+1$ configuration. We compute the asymptotic behaviour of the transition time and describe the typical tube of trajectories during the transition. We show that, unexpectedly, the mechanism of transition changes abruptly when the line $ h = 2 \lambda$ is crossed. } } \par \bigskip {\bf Keywords : Blume-Capel model, stochastic dynamics, metastability, nucleation.} \vfill\eject \numsec=1\numfor=1 {\bf Section 1. Introduction.} \par Metastability is a relevant phenomenon for thermodynamic systems close to a first order phase transition.\par Let us start from a given pure equilibrium phase in a suitable region of the phase diagram and change the thermodynamic parameters to values corresponding to a different equilibrium phase; then, in particular experimental situations, the system, instead of undergoing a phase transition, can still remain in a ``wrong" equilibrium, far from the ``true" one but actually close to what the equilibrium would be at the other side of the transition. This apparent equilibrium, often called ``metastable state", persists untill an external perturbation or some spontaneous fluctuation leads the system to the stable equilibrium.\par For a general revue on metastability with particular attention to rigorous results see [PL1],[PL2].\par There are strong arguments leading to the conclusion that neither metastability can be included in the scheme of Gibbsian formalism, which is confined to the description of the genuine stable equilibrium states (see [LR]); nor it can be directly described using extrapolation beyond the condensation point, because of the presence of an essential singularity of the free energy (see the fundamental result due to Isakov [I]).\par Metastability is a genuine dynamical phenomenon. Its description on one side has a basic importance from the point of view of fundamental physics; on the other side it poses interesting new mathematical problems ([CGOV], [OS1], [OS2]).\par Since a general approach to non-equilibrium statistical mechanics is still missing, a crucial role is played by particular models of microscopic dynamics. It is remarkable that, quite recently, rigorous results have been deduced in this field by analyzing, in particular, the geometry of the condensation nuclei as well as the possible coalescence between droplets. Notice that these aspects were totally absent in previous theories like the so called classical theory of nucleation (see [PL1]). \par In the recent years many progresses have been made in the understanding of the phenomenon of metastability in the framework of Glauber dynamics. By Glauber dynamics we mean a stochastic time evolution of a lattice spin system (in continuous or discrete time) whose elementary process is a single spin change and which is {\it reversible} (namely it satisfies the detailed balance condition) with respect to the Gibbs measure corresponding to a given hamiltonian. There is a certain freedom in chosing a particular dynamics satisfying the above mentioned requirements. A typical choice, that actually we will make in the present paper, is called ``Metropolis dynamics" (see Eq. 2.6 below).\par The case of standard Ising model (spin $+1$ or $-1$, ferromagnetic nearest neighbour interaction), often referred to as {\it Stochastic Ising model} or {\it Kinetic Ising model}, has been analyzed, in two dimensions, in [MOS] in connection to relaxation to equilibrium for arbitrary large (and even infinite) systems close to the first order phase transition.\par A quite complete treatment appeared in [NS1] and [NS2] where J. Neves and R. Schonmann analyzed, in the framework of the ``pathwise approach to metastability" introduced some years ago in [CGOV], the phenomenon of nucleation for large but finite volume and small magnetic field in the zero temperature limit.\par In [S1] R. Schonmann, using an argument based on reversibility, described in detail the typical escape paths.\par Other asymptotic regimes, very interesting from a physical point of view and mathematically much more complicated, are considered in [S2], [SS].\par In the same asymptotic regime as in [NS1], different hamiltonians have been considered in [KO1], [KO2] where it has been shown that the typical path followed during the growth of the stable phase in general are not of Wulff type. Here by Wulff (shape) we mean equilibrium shape of a droplet at zero temperature namely the shape minimizing the surface energy for fixed volume. This non-Wulff growth seems to be an interesting phenomenon in the description of crystal growth.\par Let us now try to explain the nature of the mathematical difficulties related to our problem. We notice that in the above mentioned asymptotic regime the behaviour is similar to the one described by Freidlin and Wentzell in their analysis of small random perturbations of dynamical systems: the system typically performs random oscillations around the local minima of the energy and sometimes it goes against the drift following some preferential ways. In particular it is interesting to characterize the typical tube of trajectories during the first excursion from the metastable to the stable equilibrium. This first excursion can be seen as an escape from a sort of generalized basin ${\cal G}$ of attraction of the metastable equilibrium. It turns out that many local equilibria are contained in ${\cal G}$ and this more general situation goes beyond the approach developed in [FW] by Freidlin and Wentzell who were able to give a full description of the typical tube of escape only for the case of a domain $D$ completely attracted by a unique stable point.\par In our more general case, as we will see, new interesting phenomena take place involving a sort of ``temporal entropy". These phenomena are taken into account in [OS1], [OS2], where a complete description of the typical tube of escape is given in general.\par For attractive short range systems the main feature of the transition appears to be the formation of a critical nucleus with suitable shape and size. This critical droplet results from a competition between the bulk energy favouring the growth and the surface energy favouring the contraction. Only for large sizes and for particular shapes will the droplet tend to grow.\par\bigskip The present paper is devoted to the study of metastability and nucleation in the framework of a dynamical Blume-Capel model. It is a two-dimensional spin system where the single spin variable can take three possible values: $-1,0,+1$. It was originally introduced to study the $He^3- He^4 $ phase transition. \par One can think of it as a system of particles with spin. The value $\s_x = 0$ of the spin at the lattice site $x$ will correspond to absence of particles (a {\it vacancy}), whereas the values $\s_x = +1, -1$ will correspond to the presence, at $x$, of a particle with spin $+1, -1$, respectively.\par The formal hamiltonian is given by: $$ H(\sigma)=J\sum_{}(\sigma_{x}-\sigma_{y})^{2}-\lambda\sum_{x} \sigma_{x}^{2}-h\sum_{x}\sigma_{x}\;\; , \Eq (1.1) $$ where $\lambda$ and $h$ are two real parameters, having the meaning of the chemical potential and the external magnetic field, respectively; $J$ is a real positive constant (ferromagnetic interaction) and $$ denotes a generic pair of nearest neighbour sites in ${\bf Z}^2$. \par In the following we will consider the system enclosed in a two-dimensional torus $\L$. Let $ \menouno,\;\zero\;$ and $ \;\piuuno $ denote the configurations with all the spins in $\L$ equal to$ \;-1,0,+1$, respectively. The structure of the set of ground states corresponding to different values of $\l$ and $h$ will be discussed in Section 2. Now we only note that it is immediate to see that for $\l = h = 0$ the configurations $ \menouno,\;\zero\;$ and $ \;\piuuno $ are the only ground states. It has been shown, using Pirogov-Sinai theory, that this phase transition persists at positive temperature $T=1/\beta$ in the thermodinamic limit (see [B], [C], [BS] and [DM]).\par We will use as dynamics the Metropolis algorithm, in which the typical time needed to overcome an energy barrier $\Delta H$ is of order $\exp (\beta \Delta H)$. It will be defined in detail in the next section. \par We are interested to the case in which $ \l$ and $h$ are very small but fixed, the volume is large and fixed and $T$ is very small; namely we move in the vicinities of the triple point $h=\l=T=0$. In particular we will consider the region $h >\l >0$ where the the most interesting phenomena take place. The stable equilibrium, namely the absolute minimum of the energy, in this case, is $\piuuno$ and we suppose to start with the system in the configuration $\menouno$. We want to describe the first excursion between $ \menouno$ and $\piuuno$. It turns out that in the above region a direct interface between pluses and minuses is unstable and its appearence and resistance are very unlikely.\par The main effect which surprisingly and unexpectedly shows up is that two different mechanisms of transition between $\menouno$ and $\piuuno$ take place for different values of the parameters $\l, h$. More precisely for $ 0 < 2\l 2\l$. To do this we exploit some general results contained in [OS1].\par The model-dependent part of the work consists in the solution of a well specified sequence of variational problems. The main difficulty is the determination of the ``minimal global saddle" between $\menouno$ and $\piuuno$ and of the set ${\cal G}$, the generalized basin of attraction of $\menouno$. From this we will single out an optimal nucleation mechanism. We will analyze the energy landscape so precisely to exclude all the other possible mechanisms of transition. In particular we will show that any form of coalescence will be depressed in probability with respect to the optimal nucleation mechanism.\par The paper is organized in the following way: Section 2 contains definitions and results. In particular we state Theorem 1 concerning the asymptotics of the escape time. In Section 3 we describe the local minima of the energy. In Section 4 we discuss supercriticality or subcriticality of droplets namely we determine their tendency to grow or shrink. In Section 5 we prove a basic result on the height of different global saddles. In Section 6 we define the set ${\cal G}$ and find the minima of the energy in its boundary $ \partial {\cal G}$. In Section 7 we describe the typical tube of trajectories during the first excursion; then, using as preliminary results the propositions contained in the previous section we conclude the proof of Theorem 1; finally we state and prove Theorem 2 which refers to the typical tube. Section 8 contains the conclusions. Appendix 1 contains an explicit proof of a useful result about recurrence properties of a general class of Markov chains. \vfill\eject \numsec=2\numfor=1 {\bf Section 2. Definitions and results.} \par We start by describing the model that we want to study. The configuration space is $$\O_{\L} = \{-1,0,+1\}^{\L}\;\; ,\Eq (2.1)$$ where $\L = \L_L $ is a two-dimensional torus (a square with periodic boundary conditions) of side $L$. \par A configuration $\s$ is a function: $$\s : \L \rightarrow \{ -1,0,+1\}\;\; .$$ The energy associated to the configuration $\s$ is given by: $$ H(\sigma)=J\sum_{\subset \L}(\sigma_{x}-\sigma_{y})^{2}-\lambda\sum_{x\in\L} \sigma_{x}^{2}-h\sum_{x\in\L}\sigma_{x}\;\; , \Eq (2.2) $$ where $$ denotes a generic pair of nearest neighbours sites in $\L$ and we suppose $0 < \lambda 0\; {\rm and} \; h>-\lambda\;\;\;\;\;\; &{\rm the \; ground \; state\; is \;\piuuno\; ;}\cr {\rm for }\;\; h<0\; {\rm and }\; h<\lambda\;\;\;\;\;\; &{\rm the \; ground \; state\; is \; \menouno\; ;}\cr {\rm for }\;\; \lambda<0\; {\rm and }\; \lambda 0:\; \piuuno,\menouno$ coexist. For $h=\l <0:\; \menouno,\zero$ coexist. For $h=-\l >0:\; \piuuno,\zero$ coexist. These results are summarized in Fig.2.1 where the coexistence lines are shown. \par\noindent \midinsert \vskip 7 truecm \par\noindent %\special{psfile="fig2_1.ps" voffset=-350 hoffset=-100} \par\noindent \vskip 1 truecm \centerline {{\smbfb Fig.2.1 }{\smb Ground states for the Blume-Capel model.}} \endinsert \par We want now to introduce a dynamics in our model. It will be a discrete time Glauber dynamics namely a Markov chain with state space given by $\O_{\L}$, where \par 1) the allowed transitions are between {\it nearest neighbour configurations } namely pairs $\x$ and $\eta$ of configurations differing only in one spin: $\x = \eta^{x, b}$, with $$\eta^{x,b}(y):=\left\{ \eqalign {\eta(y&)\;\;\;\;\;\;\;\forall y\in\L\;,\; y\not=x\cr b\;&\;\;\;\;\;\;\;\; {\rm for }\;\; y=x} \right. \;\; ,\Eq (2.3)$$ with $b\in \{ -1,0,+1\}$. \par 2) It is {\it reversible} w.r.t. the Gibbs measure $\m_{\L}$ for the Blume-Capel model; namely the transition probabilities $P(\s, \s')$ of the chain satisfy: $$ \m_{\L} (\s) P(\s, \s')\; = \; \m_{\L} (\s') P(\s', \s)\;\; . \Eq (2.4) $$ Our choice is the so called {Metropolis algorithm} where the transition probabilities, for pairs of different configurations $\s, \eta$, are defined as $$P(\sigma ,\eta):=\left\{ \matrix { {1\over 2|\L|} e^{-\beta [H(\eta)-H(\sigma)]^{+}}&\sigma ,\eta\; {\rm nearest \; neighbours}\cr 0&{\rm otherwise}\cr}\right.\;\; ,\Eq (2.5)$$ where: $$a^{+}:=\left\{ \matrix { a&{\rm if}\;a\geq 0\cr 0&{\rm if}\;a<0\cr}\right. \;\;\forall a\in {\bf R}\;\; .\Eq (2.6)$$ \par The {\it space of trajectories} of the process is $$\Xi :=\bigl(\O_{\L} \bigr)^{{\bf N}}\;\; .$$ An element in $\Xi$ is denoted by $\omega$; it is a function $$\omega:{\bf N} \to \O_{\L}.$$ We often write $\omega=\sigma_0,\sigma_1,\dots,\sigma_t,\dots$. \par We will call {\it path} an {\it allowed} trajectory namely: $\omega=\sigma_0,\sigma_1,\dots,\sigma_t,\dots$ is a path iff $\sigma_j$ and $\sigma_{j+1}\; \forall j$ are {\it connected} in the sense that $\sigma_{j+1} = \sigma_j^{ x,b}$ for some $x\in\L$ and $b\in\{ -1,0,+1\}$. We use the notation $\o :\s \rightarrow \h$ to denote a path $\o$ joining $\s$ to $\h$. \par A path $\o=\s_0,\s_1,\dots ,\s_n$ is called {\it downhill} ({\it uphill}) iff $H(\s_{j+1})\le H(\s_j)$ ($H(\s_{j+1})\ge H(\s_j)$) $\forall j=0,1,\dots ,n-1$. We will use the convention that a downhill path can (and will) end only in a local minimum. \par A set $Q$ of configurations, $Q \in\O_{\L}$, is said to be {\it connected} iff for every pair of configurations $\s,\h\in Q$, $\exists$ a path $\o :\s \rightarrow\h$ such that $\o\subset Q$. \par We say that a configuration $\s$ is {\it downhill connected} to $\h$ iff there exists a downhill path $\o :\s\rightarrow\h$. \par We will denote by $M$ the set of all the locally stable configurations namely the set of all the local minima of the energy. More precisely: $\s\in M$ iff for every $x\in\L ,\; b\in \{ -1,0,+1\}$ the corresponding increment in energy, given by $$\D _{x,b} H(\s):= H(\s^{ x,b}) - H(\s) \Eq (2.6')$$ is positive. \par It is easy to see that in our model with the choice of the paremeters $J,h,\l$ that we have made, the quantity $\D _{x,b} H(\s)$ will be always non-zero and this somehow simplifies some arguments. \par Given $Q\subset \O_{\L}$ we define the (outer) boundary $\partial Q$ of $Q$ as the set: $$ \partial Q := \{\s \not\in Q : \exists \s' \in Q : P(\s',\s) >0\}\;\; , $$ namely $$ \partial Q := \{\s \not\in Q :\exists x\in \L , \; b \in \{ -1, 0, +1 \} \;\hbox { such that}\; \s' = \s^{x,b}\in Q\}\;\; .\Eq (2.6'') $$ We denote by $U=U(Q)$ the set of all the minima of the energy on the boundary $\partial Q $ of $Q$: $$ U(Q) := \{ \z \in \partial Q : \min _{ \s \in \partial Q } H(\s) = H(\z) \} \Eq(2.6''') $$ and we define $H(U(Q)):=H(\xi)$ with $\xi\in U(Q)$. \par We denote by $F=F(Q)$ the set of all minima of the energy on $ Q $: $$ F(Q) := \{ \z\in Q : \min _{ \s\in Q } H(\s) = H(\z) \}\;\; . \Eq(2.6 '''') $$ Given a stable state $\s\in M$ i.e. a local minimum for the energy, we define the following {\it basins} for $\s$:\par \noindent i) the {\it wide basin of attraction of } $\s$ : $$ \hat B (\s) := \{ \z : \exists \; \hbox {downhill path } \o : \z \to \s\}\;\; ; \Eq(2.6a) $$ \par\noindent ii) the {\it basin of attraction of } $\s$ given by: $$ B(\s) := \{ \z : \hbox {every downhill path starting from }\z \; \hbox {ends in} \; \s\}\;\; , \Eq(2.6b) $$ $B(\s) $ can be seen as the usual basin of attraction of $\s$ with respect to the $\b = \infty$ dynamics. \par \noindent iii) $\bar B(\s) =$ the {\it strict basin or attraction of } $\s$ given by :\par $$ \bar B(\s) := \{ \z \in B(\s) \; : \; H(\z) < H(U( B(\s)))\}\;\; . \Eq(2.6c) $$ \par We introduce now the useful the notion of cycle. A connected set $A$ which satisfies: $$ \max _ {\s\in A} H(\s) = \bar H < \min _ {\z \in \partial A } H(\z) = H(U(A)) $$ is called {\it cycle}. Notice that every local minimum for the energy is a (trivial) cycle. \par The following simple properties of the cycles are true. Their proof is immediate (see, for instance [OS1]). \par\noindent \item{1.} Given a state $\bar \s \in \O_{\L} $ and a real number $c$ the set of all $\s$'s connected to $\bar\s $ by paths with energy always below $c$ either coincides with $\O_{\L}$ or it is a cycle $A$ with $$ H(U(A)) \geq c\;\; . $$ \item{2.} Given two cycles $A_1,\; A_2$, either i) $ A_1 \cap A_2 = \emptyset$ or ii) $A_1 \subset A_2 $ or, viceversa, $A_2 \subset A_1 $ . \par We give now some more definitions: a cycle $A$ for which there exists $\h^*\in U(A)$ downhill connected to some point $\s$ in $A^c$, is called {\it transient}; given a transient cycle $A$ the points $\h^*$ downhill connected to $A^c$ are called {\it minimal saddles}. The set of all the minimal saddles of a transient cycle A is denoted by ${\cal S}(A)$. \par A transient cycle $A$ such that $ \exists \; \bar \s \not \in A $ with $H(\bar \s) < H(F(A))$, there exists $\h^* \in {\cal S}(A)$ and a path $ \o : \h^* \to \bar \s$ {\it below } $\h^*$ (namely $\forall\s \in \o : H(\s) < H(\h^*)$), is called {\it metastable}. \par For each pair of states $\s,\h\in \O_{\L}$ we define their minimal saddle $ {\cal S}(x,y)$ as the set of states corresponding to the solution of the following minimax problem: let, for any path $\o$ $$ \hat H(\o)\; := \max _{ \z\in \o} H(\z), \;\; \;\;\;\;\;\;\bar H_{\s,\h} \;:=\;\min_{\o : \s \to \h} \hat H (\o)\;\; , $$ find $$ {\cal S}(\s,\h):= \{ \z:\; H(\z) = \bar H_{\s,\h} ; \; \exists \;\o : \;\s \to \h ,\; \o \ni \z,\;\hat H(\o) = \bar H_{\s,\h} \}\;\; . $$ One immediately verifies that a strict basin of attraction of a local minimum is a transient cycle. This case corresponds to a ``one well" structure. More general cases involve the presence of ``internal saddles" and correspond to a ``several wells" situation. \par Given any set of configurations $A\subset \O_{\L}$, we use $\tau_A$ to denote the {\it first hitting time} to $A$: $$ \tau_A :=\inf\{t\geq 0 \;: \sigma_t\in A\}\;\; . \Eq (2.7) $$ We use $P_\eta(\cdot)$ to denote the probability distribution over the process starting at $t=0$ from the configuration $\eta$.\par We are interested in dynamics at very low temperatures. Namely, we will discuss the asymptotic behaviour, in the limit $\beta\to\infty$, of typical paths of the first escape from $-\underline 1$ to $+\underline 1$ for fixed $h, \; \l $ and $\L$. \par Let us now better clarify the asymptotic regime in which we will operate: the volume $|\L|$, the magnetic field $h$ and the chemical potential $\l$ are fixed and we consider asymptotic estimates for $\b$ very large. This regime was studied in the case of standard Ising model in 2D by J. Neves and R. Schonmann in [NS1] where the point of view of the {\it pathwise approach to metastability}, introduced in [CGOV], was assumed. \par One can think, for instance, to take $\l$ very small, $h=a\l$, $a$ fixed positive number, $|\L|$ of order, say, of ${1\over h^2}$ and $\b$ of order $ 1\over h^5$; physically this corresponds to a regime in which, at the equilibrium, the energy dominates w.r.t. the entropy.\par In the above described situation the qualitative behaviour of our stochastic time evolution can be described as follows: the system will spend the majority of the time in the local minima of the energy. Sometimes it escapes from them but there is always a natural tendency to follow a downhill path and an occasional, random and rather unprobable, uphill move.\par An important role will be played by the saddles separating different ``basins of attraction'' (or generalized basins of attraction, see below) w.r.t. the $\b =\infty$ dynamics. \par We will see that the local minima will correspond to particular geometric shapes that will be called {\it plurirectangles} (see Fig.3.8); we will analyze, in particular, the special saddles between ``contiguous" local minima (see Lemma 5.1). \par A {\it global saddle point} is any configuration $$\bar\sigma \; \in {\cal S}(- \underline 1,+ \underline 1)\;\; .$$ In Section 6 we will see that the set of these global minima ${\cal P}$ are substantially different according to the values of the parameters $\l$ and $h$. \par\noindent 1) For $ h < 2\l$ we distinguish two cases:\par a) if $\d := l^* - { 2J -(h-\l)\over h} < { h +\l \over 2h}$, ${\cal P}$ is of the form ${\cal P}_{1,a}$ given in Fig.5.1 (the two critical lengths $l^*$ and $M^*$ are defined in $(3.12)$ and $(3.15)$); namely it contains a ``droplet" with external rectangle given by a square of side $l^*+2$; the internal shape given by a rectangle with sides $ l^*,\; l^* -1$, at a distance one from the external rectangle and with a unit square protuberance attached to the longest ``free" side; the internal shape is full of pluses, the spins lying outside to the exterior rectangle are minuses; finally between the interior shape and the external rectangle there are zeroes.\par b) If $\d > { h +\l \over 2h}$, ${\cal P}$ is of the form ${\cal P}_{1,b }$ depicted in Fig.5.1. ${\cal P}_{1,b }$ is similar to ${\cal P}_{1,a }$ but now the external rectangle has sides $l^*+1,\; l^* +2$ and internally we have a square with sides $ l^*-1$ with a unit square protuberance attached to the shortest ``free" side. \par \noindent 2) For $ h > 2\l$, ${\cal P}$ is of the form ${\cal P}_2$ given in Fig.5.1; namely it is given by a rectangle of sides $M^*$ and $M^*-1$, with a unit square protuberance attached to one of its longest sides, full of zeroes in a ``sea" of minuses. \par We set: $$\G \; := \; H ( {\cal P}) - H ( \menouno)\;\; . \Eq (2.10)$$ Let us now summarize our main results.\par We shall prove that the first excursion from $-\underline 1$ to $+\underline 1$ typically passes through a configuration from ${\cal P}$ and the time needed for this to happen is of the order $\exp(\beta\Gamma)$; this is the content of Theorem 1 that we are now going to state. \par Theorem 2 will characterize the typical trajectories of the first excursion. The proof of Theorems 1,2 and even the statement of Theorem 2 will need many more definitions and propositions. For this reasons they will be postponed to Section 7. \par Theorem 1 is based, in particular, on Propositions 4.1, 4.2 and 4.3 given in Section 4. These propositions refer to the tendency of a given minimum $\eta$ of the energy to evolve towards $ \piuuno$ or to $\menouno$ namely they establish under which conditions a {\it droplet} is {\it supercritical} or not.\par It will be crucial to introduce a sort of generalized basin of attraction of $\menouno$. Indeed we will reduce the proof of Theorem 1 to finding a certain set ${\cal G}$ of configurations satisfying suitable properties. In order to explicitly construct this set ${\cal G}$ we will need the results contained in Propositions 4.1, 4.2 and 4.3. This construction will be achieved in Section 6. %%%%%%%%%%%%%%%%%Theorem 1 \vskip 0.35 truecm \noindent {\bf Theorem 1.}\par\noindent Let $\bar\tau_{-\underline 1}$ be the last instant in which $\sigma_t= -\underline 1$ before $\tau_{+\underline 1}$: $$ \bar\tau _{-\underline 1} :=\max \{t<\tau_{+\underline 1}: \sigma_t=-\underline 1\}\;\; .\Eq (2.11) $$ Let $$ \bar\tau_{\cal P} :=\min\{t> \bar\tau_{-\underline 1} : \sigma_t = \cal P\}\;\; ; \Eq (2.12) $$ for every $\varepsilon > 0$: \par\noindent \vskip 0.35 truecm i) $$ \lim_{\beta\to\infty}P_{-\underline 1}(\bar\tau_{\cal P} <\tau_{+\underline 1})=1\;\; ; \Eq (2.13) $$ \par ii) $$ \lim_{\beta\to\infty}P_{-\underline 1}(\exp[\beta(\Gamma-\varepsilon)] <\tau_{+\underline 1}<\exp[\beta(\Gamma+\varepsilon)])=1\;\; .\Eq (2.14) $$ %%%%%%%%%%%%%%%%%%%End of Theorem 1 \vfill\eject \numsec=3\numfor=1 {\bf Section 3. Local minima of the hamiltonian $H(\s)$.} \par In this section we want to analyze the geometrical structure of the local minima of the energy. \par For any configuration $\s\in\O_\L$ we denote by $c^{+}(\s)$, $c^{-}(\s)$ and $c^{0}(\s)$ the union of all closed unit squares centered at sites $x\in\L$ with $\s(x)$ respectivly equal to $+1,-1$ and $0$. $c^{+}(\s)$, $c^{-}(\s)$ and $c^{0}(\s)$ decompose into maximal connected components $c^{+,0,-}_j, \; j=1, \dots , k^{+,0,-}$.\par The centers of $c^{+,0,-}_j$ form a $\star$--cluster in the sense of sites percolation, namely they are maximally connected components in the sense of the next nearest neighbours. The $c^{+,0,-}_j$ will be simply called {\it clusters}. \par To any such $c^{+,0,-}_j$ we assign its {\it rectangular envelope} defined as the minimal closed rectangle $R(c^{\pm ,0}_{j})$ containing it; if none of the rectangles $R(c^{+,0}_{j})$ is winding around the torus, we call the corresponding configuration {\it acceptable}. \par Let $\s$ be an acceptable configuration, we denote by $\g^{+,0}_{j}$ the boundary of $c^{+,0}_{j}\;\;\forall j\in\{1,...,k^{+,0}\}$; the internal component ${\check\g^{+,0}_{j}}$ of the boundary is defined as follows: let {\it s} be a unit segment of the dual lattice ${\bf Z}^{2}+({1\over 2},{1\over 2})$ belonging to $\g^{+,0}_{j}$, we say that $s\in{\check\g^{+,0}_{j}}$ if and only if all the paths joining nearest neighbour sites of $\L$ and starting from the site adjacent to {\it s} and not in $c^{+,0}_{j}$, necessarily reach a site in $c^{+,0}_{j}$ before touching the cluster $c^{-}_{j}$ winding around the torus. The external component ${\hat\g^{+,0}_{j}}$ of the boundary of $c^{+,0}_{j}$ is defined as $\g^{+,0}_{j}\setminus{\check\g^{+,0}_{j}}$. Of course ${\check\g^{+,0}_{j}}$ can be empty. \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~~0~~~$\phantom {\left\{\matrix {0& {\char45}6J{\char45}(h{\char45}\l)\cr +& {\char45}4J{\char45}2h\cr}\right.}$~~~~~~~~~~~~~~~~0} \hbox{\amgr ~~~~~~~~~~~+~{\char45}~0~~~$\left\{\matrix {0& {\char45}6J{\char45}(h{\char45}\l)\cr +& {\char45}4J{\char45}2h\cr}\right.$~~~~~~~~~~~~+~{\char45}~0~~~$\left\{\matrix {0& {\char45}8J{\char45}(h{\char45}\l)\cr +& {\char45}8J{\char45}2h\cr}\right.$} \hbox{\amgr ~~~~~~~~~~~~~0~~~$\phantom {\left\{\matrix {0& {\char45}6J{\char45}(h{\char45}\l)\cr +& {\char45}4J{\char45}2h\cr}\right.}$~~~~~~~~~~~~~~~~+} \hbox{\amgr } \hbox{\amgr } \hbox{\amgr ~~~~~~~~~~~~~0~~~$\phantom {\left\{\matrix {0& {\char45}4J{\char45}(h{\char45}\l)\cr +& {\char45}3J{\char45}2h\cr}\right.}$~~~~~~~~~~~~~~~~0} \hbox{\amgr ~~~~~~~~~~~+~{\char45}~0~~~$\left\{\matrix {0& {\char45}4J{\char45}(h{\char45}\l)\cr +& {\char45}3J{\char45}2h\cr}\right.$~~~~~~~~~~~~+~{\char45}~+~~~$\left\{\matrix {0& {\char45}10J{\char45}(h{\char45}\l)\cr +& {\char45}12J{\char45}2h\cr}\right.$} \hbox{\amgr ~~~~~~~~~~~~~{\char45}~~~$\phantom {\left\{\matrix {0& {\char45}4J{\char45}(h{\char45}\l)\cr +& {\char45}3J{\char45}2h\cr}\right.}$~~~~~~~~~~~~~~~~+} \hbox{\amgr } \hbox{\amgr } \hbox{\amgr ~~~~~~~~~~~~~0~~~$\phantom {\left\{\matrix {0& {\char45}6J{\char45}(h{\char45}\l)\cr +& {\char45}4J{\char45}2h\cr}\right.}$~~~~~~~~~~~~~~~~0} \hbox{\amgr ~~~~~~~~~~~+~{\char45}~+~~~$\left\{\matrix {0& {\char45}6J{\char45}(h{\char45}\l)\cr +& {\char45}4J{\char45}2h\cr}\right.$~~~~~~~~~~~~+~{\char45}~{\char45}~~~$\left\{\matrix {0& {\char45}2J{\char45}(h{\char45}\l)\cr +& +4J{\char45}2h\cr}\right.$} \hbox{\amgr ~~~~~~~~~~~~~{\char45}~~~$\phantom {\left\{\matrix {0& {\char45}6J{\char45}(h{\char45}\l)\cr +& {\char45}4J{\char45}2h\cr}\right.}$~~~~~~~~~~~~~~~~{\char45}} \hbox{\amgr } \hbox{\amgr } \hbox{\amgr ~~~~~~~~~~~~~+~~~$\phantom {\left\{\matrix {0& {\char45}12J{\char45}(h{\char45}\l)\cr +& {\char45}16J{\char45}2h\cr}\right.}$~~~~~~~~~~~~~~~+} \hbox{\amgr ~~~~~~~~~~~+~{\char45}~+~~~$\left\{\matrix {0& {\char45}12J{\char45}(h{\char45}\l)\cr +& {\char45}16J{\char45}2h\cr}\right.$~~~~~~~~~~~+~{\char45}~+~~~$\left\{\matrix {0& {\char45}8J{\char45}(h{\char45}\l)\cr +& {\char45}8J{\char45}2h\cr}\right.$} \hbox{\amgr ~~~~~~~~~~~~~+~~~$\phantom {\left\{\matrix {0& {\char45}12J{\char45}(h{\char45}\l)\cr +& {\char45}16J{\char45}2h\cr}\right.}$~~~~~~~~~~~~~~~{\char45}} \hbox{\amgr } \hbox{\amgr } \hbox{\amgr ~~~~~~~~~~~~~+~~~$\phantom {\left\{\matrix {0& {\char45}4J{\char45}(h{\char45}\l)\cr +& {\char45}2h\cr}\right.}$~~~~~~~~~~~~~~~~{\char45}} \hbox{\amgr ~~~~~~~~~~~+~{\char45}~{\char45}~~~$\left\{\matrix {0& {\char45}4J{\char45}(h{\char45}\l)\cr +& {\char45}2h\cr}\right.$~~~~~~~~~~~~+~{\char45}~{\char45}~~~$\left\{\matrix {0& {\char45}(h{\char45}\l)\cr +& +8J{\char45}2h\cr}\right.$} \hbox{\amgr ~~~~~~~~~~~~~{\char45}~~~$\phantom {\left\{\matrix {0& {\char45}4J{\char45}(h{\char45}\l)\cr +& {\char45}2h\cr}\right.}$~~~~~~~~~~~~~~~~{\char45}}} \vskip 0.35 truecm \par\noindent \centerline {\smbfb Fig.3.1} \endinsert \par In order to construct the local minima of the hamiltonian we first prove that direct $+-$ interfaces cannot exist in such configurations; in Fig.3.1 we analyze the interaction of a minus spin with its neighbouring sites. We examine all the possible cases and we show that it is always possible to construct a lower energy configuration by changing the minus spin adjacent to the interface. \par Let $\s$ be an acceptable configuration such that there exists only one cluster of $0$ spins $c^0$ and no plus spins; it can be proved that $$\sigma\;{\rm is\; a\; local\; minimum\; of\;}H(\s)\;\Longleftrightarrow\; \left\{\eqalign { \g^{0}&={\hat\g^{0}}\;{\rm is\; a\; rectangle\; whose}\cr {\rm si}&{\rm des\; are\; longer\; than\; two}\cr}\right.\;\; .\Eq (3.1)$$ Indeed, if $\s$ is a local minimum and there exists a minus spin inside the cluster $c^0$, then, as a consequence of the fact that $c^0$ does not wind around the torus, one has that necessarily it must exist at least one minus spin with at least two nearest neighbour sites occupied by $0$ spins (see Fig.3.2). \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char2}-----{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~0~~{\char0}~~~{\char2}--{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~--{\char4}~~~~~{\char0}~~~{\char0}~~{\char5}--{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char5}---{\char4}~~0~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char45}~{\char45}~{\char45}~{\char45}~---------{\char3}~~{\char5}-{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-~{\char0}~0~~{\char0}}} \vskip 0.35 truecm \par\noindent \centerline {\smbfb Fig.3.2} \par\noindent This minus spin can be changed into $+$ or $0$ by obtaining, in this way, a lower energy configuration, as shown in Fig.3.3; this is an absurd. \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~~0~~~$\phantom {\left\{\matrix {0& {\char45}4J{\char45}(h{\char45}\l)\cr +& {\char45}2h\cr}\right.}$~~~~~~~~~~~~~~~~0} \hbox{\amgr ~~~~~~~~~~~0~{\char45}~0~~~$\left\{\matrix {0& {\char45}4J{\char45}(h{\char45}\l)\cr +& {\char45}2h\cr}\right.$~~~~~~~~~~~~{\char45}~{\char45}~0~~~$\left\{\matrix {0& {\char45}(h{\char45}\l)\cr +& +8J{\char45}2h\cr}\right.$} \hbox{\amgr ~~~~~~~~~~~~~0~~~$\phantom {\left\{\matrix {0& {\char45}4J{\char45}(h{\char45}\l)\cr +& {\char45}2h\cr}\right.}$~~~~~~~~~~~~~~~~+} \hbox{\amgr } \hbox{\amgr } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~0} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~{\char45}~{\char45}~0~~~$\left\{\matrix {0& {\char45}2J{\char45}(h{\char45}\l)\cr +& +4J{\char45}2h\cr}\right.$} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~0}} \vskip 0.35 truecm \par\noindent \centerline {\smbfb Fig.3.3} \par\noindent We can conclude that no minus spins can be inside $c^0$, that is ${\check\g^0}=\{\emptyset\}$. In a similar way it can be proved that ${\hat\g^0}$ is a rectangle and its sides are longer than two. \par The proof of the implication $\Leftarrow$ is in Fig.3.4, where it is shown that all the possible nearest neighbour configurations of $\s$ are at higher energy; in Fig.3.4 the modified spin is represented by a unit empty square. \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~{\char2}------------------------{\char3}} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~-} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~0~~~~~~~~~~~~~{\char0}~{\char2}-{\char3}~~~~~~~~~~$\left\{ \matrix { 0& +4J{\char45}(h{\char45}\l)\cr +& +16J{\char45}2h\cr}\right.$} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char5}-{\char4}} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~{\char5}------------------------{\char4}} \hbox{\amgr } \hbox{\amgr } \hbox{\amgr ~~~~~~~~{\char2}------------------------{\char3}} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~{\char15}-{\char3}~~~~~~~~~~~~$\left\{ \matrix { 0& +2J{\char45}(h{\char45}\l)\cr +& +12J{\char45}2h\cr}\right.$} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~{\char15}-{\char4}} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~{\char5}------------------------{\char4}} \hbox{\amgr } \hbox{\amgr } \hbox{\amgr ~~~~~~~~{\char2}----------------------{\char18}-{\char3}} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~{\char5}-{\char16}~~~~~~~~} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~$\left\{ \matrix { {\char45}& +(h{\char45}\l)\cr +& +8J{\char45}(h+\l)\cr}\right.$} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~{\char5}------------------------{\char4}} \hbox{\amgr ~} \hbox{\amgr } \hbox{\amgr ~~~~~~~~{\char2}------------------------{\char3}} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~{\char2}-{\char16}~~~~~~~~~~~~~~~$\left\{ \matrix { {\char45}& +2J+(h{\char45}\l)\cr +& +6J{\char45}(h+\l)\cr}\right.$} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~{\char5}-{\char16}} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~{\char5}------------------------{\char4}} \hbox{\amgr } \hbox{\amgr } \hbox{\amgr ~~~~~~~~{\char2}------------------------{\char3}} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~{\char2}-{\char3}~{\char0}~~~~~~~~~~~~~~~$\left\{ \matrix { {\char45}& +4J+(h{\char45}\l)\cr +& +4J{\char45}(h+\l)\cr}\right.$} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~{\char5}-{\char4}~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~{\char5}------------------------{\char4}}} \vskip 0.35 truecm \par\noindent \centerline {\smbfb Fig.3.4} \endinsert \par Now let $\s$ be an acceptable configuration such that the following conditions are satisfied: there exists just one cluster $c^0$ of $0$ spins touching $c^-$, ${\hat\g^0}$ is a rectangle, no minus spin is inside clusters of $0$ spins; all plus spins are in the cluster $c^+$ and ${\hat\g^+}={\check\g^0}$ (see Fig.3.5). \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char2}---------------------------------------{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~{\char2}-{\char3}~~~0~~{\char2}-----{\char3}~~~~~~{\char2}---{\char3}~~~~~~~~{\char0}% } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~{\char0}~{\char5}------{\char4}~~+~~{\char5}-{\char3}~~~~{\char0}~~~{\char5}-----% {\char3}~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~{\char0}~~~~~~~~~~~~~~~~{\char5}----{\char4}~~~~~+~~~{\char0}~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~{\char0}~~~~~{\char2}----{\char3}~~~~~~~~~~~~~{\char2}----{\char3}~{\char0}~~{\char0}% } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char0}~~{\char2}-{\char4}~~~{\char2}-{\char4}~~~~{\char5}----{\char3}~~~{\char2}----{\char4}% ~~~~{\char0}~{\char0}~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char0}~~{\char0}~~~~~{\char0}~~~~0~~~~~~{\char0}~~~{\char5}-{\char3}~~~~{\char45}~~{\char0}~% {\char0}~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char0}~~{\char5}---{\char3}~{\char5}---{\char3}~~~0~~~{\char0}~~~~~{\char0}~{\char45}~~~{\char2}% -{\char4}~{\char0}~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~{\char0}~~~~~{\char0}~~~~~{\char2}-{\char4}~~~~~{\char5}-----{\char4}~~~{\char0}% ~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~{\char5}--{\char3}~~{\char5}-{\char3}~~~{\char0}~~+~~~~~~~~+~~~{\char2}-{\char4}% ~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~{\char0}~~~~{\char5}---{\char4}~~~~{\char2}----{\char3}~~~{\char2}-{\char4}~% ~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~0~~~~~{\char0}~~~~~~~~~~~~~{\char0}~~~~{\char5}---{\char4}~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~{\char5}----{\char3}~~~~~~{\char2}-{\char4}~~~~~~~~~~0~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~{\char5}------{\char4}~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char5}---------------------------------------{\char4}}} \vskip 0.35 truecm \par\noindent \centerline {\smbfb Fig.3.5} \endinsert \par\noindent With arguments similar to the ones used before, it can be proved that $$\sigma\;{\rm is\; a\; local\; minimum\; of\;}H(\s)\;\Longleftrightarrow\; \left\{\eqalign { \g^+&={\hat \g^{+}}\;{\rm is\; a\; rectangle\; whose}\cr {\rm si}&{\rm des\; are\; longer\; than\; two}\cr}\right.\;\; .\Eq (3.2)$$ In the proof it is crucial that the energy of a configuration can be lowered by properly changing a $0$ spin having at most two zero spins and no minus spins among its nearest neighbour sites; all the possible situations are shown in Fig.3.6. \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~~+~~~$\phantom {\left\{\matrix {0& 8J+(h{\char45}\l)\cr +& {\char45}(h+\l)\cr}\right.}$~~~~~~~~~~~~~~~~+} \hbox{\amgr ~~~~~~~~~~~0~0~+~~~$\left\{\matrix {{\char45}& 8J+(h{\char45}\l)\cr +& {\char45}(h+\l)\cr}\right.$~~~~~~~~~~~~0~0~+~~~$\left\{\matrix {{\char45}& 10J+(h{\char45}\l)\cr +& {\char45}2J{\char45}(h+\l)\cr}\right.$} \hbox{\amgr ~~~~~~~~~~~~~0~~~$\phantom {\left\{\matrix {{\char45}& 8J+(h{\char45}\l)\cr +& {\char45}(h+\l)\cr}\right.}$~~~~~~~~~~~~~~~~+} \hbox{\amgr } \hbox{\amgr } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~+} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~+~0~+~~~$\left\{\matrix {{\char45}& 12J+(h{\char45}\l)\cr +& {\char45}4J{\char45}(h+\l)\cr}\right.$} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~+}} \vskip 0.35 truecm \par\noindent \centerline {\smbfb Fig.3.6} \par Hence we have proved that configurations like the one in Fig.3.7 are local minima of $H(\s)$; these configurations are called {\it birectangles} and are denoted by the symbol $R(L_{1},L_{2};M_{1},M_{2})$ where $$\left\{ \matrix { M_{1}\geq L_{1}+2,M_{2}\geq L_{2}+2 &\; {\rm if}\;\; L_{1},L_{2}\geq 2\cr M_{1},M_{2}\geq 2 &\; {\rm if}\;\; L_{1}=L_{2}=0\cr}\right.\;\; .\Eq (3.3)$$ \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char2}-------------------------------{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~$L_{1}\;\,\,$~~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char2}-------{\char3}~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~{\char0}~$M_{2}$} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~{\char0}~$L_{2}\;\, \,$~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char5}-------{\char4}~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char5}-------------------------------{\char4}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$M_{1}$}} \vskip 0.35 truecm \par\noindent \centerline {\smbfb Fig.3.7} \endinsert \par It is easy to understand that the most general local minimum of $H(\s)$ is not a birectangle, but, rather, a more complicate configuration that we call family of {\it plurirectangle} (see Fig.3.8). It is an acceptable configuration satisfying the following conditions: \vskip 0.35 truecm \itemitem{i)} there are $k^0$ clusters $c^0_1,...,c^0_{k_0}$ of $0$ spin touching $c^-$; \itemitem{ii)} ${\hat\g^0_1},...,{\hat\g^0_{k_0}}$ are non interacting rectangles whose sides are longer than two; \itemitem{iii)} in every cluster $c^0_j$ there are $k^+_{j}$ clusters $c^+_1,...,c^+_{k^+_j}$ of +1 spins; \itemitem{iv)} $\forall j\in\{1,...,k^{0}\}\;$ ${\hat\g^+_{j,1}},...,{\hat\g^+_{j,k^+_j}}$ are non interacting rectangles whose sides are longer than two. \vskip 0.35 truecm \par\noindent We have a single plurirectangle when $k^0=1$. \par We have used, above, the geometric notion of interacting rectangles: given two rectangles $R_1$ and $R_2$ with boundaries on the dual lattice ${\bf Z}^{2}+({1\over 2},{1\over 2})$, we say that they {\it interact} if and only if one of the two following conditions occurs: \vskip 0.35 truecm \itemitem{i)} their boundaries intersect; \itemitem{ii)} there exists a unit square centered at some lattice site such that two of its edges are opposite and lie respectively on the boundaries of $R_1$ and $R_2$. \vskip 0.35 truecm \par \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~~~{\char2}----------------------------------------------------------{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~{\char2}---------------------{\char3}~~~~~~~~~~~~~~~~~~~~~{\char45}~~~~~~{\char0}% } \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~{\char0}~{\char2}-------{\char3}~~~~~0~~~~~{\char0}~~~{\char2}-----------{\char3}% ~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~{\char0}~{\char0}~~~~~~~{\char0}~~{\char2}----{\char3}~~~{\char0}~~~{\char0}~~% ~~~~~~~~~{\char0}~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~{\char0}~{\char0}~~~~~~~{\char0}~~{\char0}~~~~{\char0}~~~{\char0}~~~{\char0}~~% ~~~~~~~~~{\char0}~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~{\char0}~{\char0}~~~+~~~{\char0}~~{\char0}~~+~{\char0}~~~{\char0}~~~{\char0}~~% ~~~0~~~~~{\char0}~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~{\char0}~{\char0}~~~~~~~{\char0}~~{\char0}~~~~{\char0}~~~{\char0}~~~{\char0}~~% ~~~~~~~~~{\char0}~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~{\char0}~{\char0}~~~~~~~{\char0}~~{\char5}----{\char4}~~~{\char0}~~~{\char5}--% ---------{\char4}~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~{\char0}~{\char0}~~~~~~~{\char0}~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~~~~% ~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~{\char0}~{\char5}-------{\char4}~{\char2}------{\char3}~~{\char0}~~~~~~~~~{\char45}% ~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~{\char0}~~~~~~~~~~~{\char0}~~~~~~{\char0}~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~~~~% ~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~{\char0}~~~~~0~~~~~{\char0}~~+~~~{\char0}~~{\char0}~~~~~~~~~{\char2}----------% -----{\char3}~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~{\char0}~~~~~~~~~~~{\char0}~~~~~~{\char0}~~{\char0}~~~~~~~~~{\char0}~~~~~~~~~~% ~~~~~{\char0}~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~{\char0}~~~~~~~~~~~{\char5}------{\char4}~~{\char0}~~~~~~~~~{\char0}~{\char2}-% ----------{\char3}~{\char0}~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~{\char5}---------------------{\char4}~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}% ~{\char0}~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}~% ~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~+~~~~~{\char0}~{\char0}~% ~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}~% ~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~{\char45}~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char5}-----------{\char4}~{\char0}% ~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~{\char0}~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char5}---------------{\char4}~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char5}----------------------------------------------------------{\char4}~~~~}} \vskip 0.35 truecm \par\noindent \centerline {\smbfb Fig.3.8} \endinsert \par We have to compute the energy of such local minima as a first step in the description of their tendency to shrink or grow of the stables clusters. \par We say that a local minimum $\s$ is {\it subcritical} if and only if $$\lim_{\beta\to\infty}P_{\s}(\tau_{\menouno}<\tau_{\piuuno})=1\;\; ; \Eq (3.3.1)$$ one of the main problems that we have to solve is to understand when a local minimum is subcritical.\par The energy of a birectangle $R(L_{1},L_{2};M_{1},M_{2})$ is $$H(R(L_{1},L_{2};M_{1},M_{2}))-H(\menouno)=$$ $$=(2M_{1}+2M_{2})J+(2L_{1}+2L_{2})J- M_{1}M_{2}(h-\l)-L_{1}L_{2}(h+\l)\;\; .\Eq (3.4)$$ The above formula can be easily generalized to the case of a general plurirectangle $\s$, characterized by the parameters $M_{1,j},M_{2,j},L_{1,j,i}\;{\rm e}\; L_{2,j,i}\;\;\forall j\in\{1,...,k^{0}\}\; {\rm and }\;\forall i\in\{1,...,k^{+}_{j}\}$, with obvious meaning of the notation. One has $$\eqalign { H(\sigma)-H(\menouno)=\sum_{j=1}^{k^{0}}&\big\{ (2M_{1,j}+2M_{2,j})J-M_{1,j}M_{2,j}(h-\l)+\cr +\sum_{i=1}^{k^{+}_{j}} [ (2L_{1,j,i}+&2L_{2,j,i})J-L_{1,j,i}L_{2,j,i}(h+\l)]\big\}\cr}\;\; .\Eq (3.5)$$ \par Now we consider a squared birectangle $Q(L,M):=R(L,L;M,M)$, whose energy $e(M,L):=H(Q(L,M))-H(\menouno)$ is given by $$e(M,L)=4MJ+4LJ-M^{2}(h-\l)-L^{2}(h+\l)\;\; .\Eq (3.6)$$ The graph of this function $ e: {\bf R}^2 \to {\bf R}$ is a paraboloid with elliptical section and downhill concavity, the coordinates of the vertex are $$M={2J\over h-\l}\;\;\;\;\;\;L={2J\over h+\l};\Eq (3.7)$$ the level curves of $e(M,L)$ are represented in Fig.3.9. \midinsert \vskip 9 truecm\noindent %\special{psfile="fig3_9.ps" voffset=-300 hoffset=-100} \centerline {\smbfb Fig.3.9} \endinsert \par Let us consider a droplet $Q(M,L)$ such that $M<{2J\over h-\l}$ and $L<{2J\over h+\l}$: if these conditions are satisfied $e(M,L)$ is an increasing function of $M$ and $L$, so we expect that this droplet will shrink. On the other hand if $M>{2J\over h-\l}$, since $e(M,L)$ is a decreasing function of $M$, we expect that the external cluster of the droplet will grow; this suggests that $M^{*}:=\left[ {2J\over h-\l}\right] +1$ is the {\it critical dimension} for the external cluster of a local minimum. After the growth of the external cluster, we look at what will happen to the internal one; with similar arguments one can convince himself that $L^{*}:=\left[ {2J\over h+\l}\right] +1$ appears to play the role of the critical dimension. Obviously these two processes of growth cannot be inverted, in fact a plus spin droplet can ``live" only inside a zero spin droplet. \par But it can also happen that the plus spin phase is reached directly, without passing through the zero spin phase; this happens if the droplet $Q(M,L)$ grows moving along the line $M=L+2$. In this case one can see that the system reaches the stable phase through a sequence of frames (picture frames). We call {\it squared frame} a birectangle $C(l,l):=R(l,l;l+2,l+2)$ with $l\ge 2$. The most general {\it frame} is a rectangular one (see Fig.3.10) $$C(l_{1},l_{2}):=R(l_{1},l_{2};l_{1}+2,l_{2}+2)\;\; ,\Eq (3.8)$$ where $l_1,l_2\ge 2$. \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char2}---------------------{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char2}-----------------{\char3}~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}0{\char0}~~~~~~~~~~~~~~~~~{\char0}0{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~+~~~~~~~~{\char0}~{\char0}~~~$l_2+2$} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char5}-----------------{\char4}~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char5}---------------------{\char4}} \hbox{\amgr } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$l_1+2$~~~~~~~~~~}} \vskip 0.35 truecm \par\noindent \centerline {\smbfb Fig.3.10 } \endinsert \par Now we consider the energy of a squared frame $e(l):=H(C(l,l)) -H(\menouno)$, using equality $\equ (3.6)$ we have $$e(l)=-2hl^{2}+l[8J-4(h-\l)]+[8J-4(h-\l)]\;\; ;\Eq (3.9)$$ the graph of this function is a concave parabola, whose vertex coordinate is $$l={2J-(h-\l)\over h}\;\; .\Eq (3.10)$$ We expect that $C(l,l)$ will grow if $l\ge l^*$, where $$l^*:=\left[ {2J-(h-\l)\over h}\right] +1\;\; ,\Eq (3.10.1)$$ otherwise it will shrink; hence $l^*$ should be the critical dimension of a squared frame. \par In order to describe the behaviour of a general birectangle $R=R(L_1,L_2;M_1,M_2)$, we must study growth and contraction mechanisms of a droplet; like in the Ising model these are mainly: growth of a (unit square) protuberance and corner erosion. But in Blume-Capel model the relevant local minima are made of two components, the internal and the external ones, and both of them can grow or shrink independently. The mechanisms of growth and contraction are explained in Fig.3.11, they corrispond to: \vskip 0.35 truecm \itemitem{1)} creation of a $+$ protuberance adjacent from the exterior to the internal rectangle; \itemitem{2)} creation of a $0$ protuberance adjacent from the exterior to the external rectangle; \itemitem{3)} erosion $(+\rightarrow 0)$ of all but one $+$ spin in a row or column of the internal rectangle; \itemitem{4)} erosion $(0\rightarrow -)$ of all but one $0$ spin in a row or column of the external rectangle. \vskip 0.35 truecm \par\noindent Their typical times are $$\eqalign { t_{1}=e^{\b[2J-(h+\l)]}\;\;\;&t_{2}=e^{\b[2J-(h-\l)]}\cr t_{3}=e^{\b(h+\l)(L-1)}\;\;\;&t_{4}=e^{\b(h-\l)(M-1)}\cr} \;\; ,\Eq (3.11)$$ where $L:=\min \{L_{1},L_{2}\}$ and $M:=\min \{M_{1},M_{2}\}$. \midinsert \vskip 0.35 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~{\char2}-----------------------{\char3}~~~~~~~~~~~~~~~{\char2}-----------------------{\char3}% } \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~{\char5}-{\char3}~~~~~~~~~~~~~~~~~~~~~{\char0}% } \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~{\char0}% } \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~{\char2}----------{\char3}~~~~{\char0}~~~~~~~~~~~~~~~|~{\char0}~~~~~{\char2}--% --------{\char3}~~~~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~{\char0}~~~~~~~~~~{\char0}~~~~{\char0}~~~~~~~~~~~~~~~~~{\char0}~~~~~{\char5}-{\char3}% ~~~~~~~~{\char0}~~~~{\char5}-{\char3}} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~{\char0}~~~~~~~~~~{\char0}~~~~{\char0}~~~~~~~~~~~~~~~|~{\char0}~~~~~~~{\char0}% ~~~~~~~~{\char0}~~~~{\char2}-{\char4}} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~{\char0}~~~~~~~~~~{\char0}~~~~{\char0}~~~~~~~~~~~~~~4~~{\char0}~~~~~|~{\char0}% ~~~~~~~~{\char0}~~~~{\char0}~2} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~{\char0}~~~~~~~~~~{\char0}~~~~{\char0}~~~~~~~~~~~~~~~|~{\char0}~~~3~~~{\char0}% ~~~~~~~~{\char5}-{\char3}~~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~{\char0}~~~~~~~~~~{\char0}~~~~{\char0}~~~~~~~~~~~~~~~~~{\char0}~~~~~|~{\char0}% ~~~~~~~~{\char2}-{\char4}~~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~{\char0}~~~~~~~~~~{\char0}~~~~{\char0}~~~~~~~~~~~~~~~|~{\char0}~~~~~~~{\char0}% ~~~~~~~~{\char0}~1~~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~{\char5}----------{\char4}~~~~{\char0}~~~~~~~~~~~~~~~~~{\char0}~~~~~{\char5}~{\char17}% --------{\char4}~~~~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~|~{\char0}~~~~~~~~~~~~~~~~~~~~~{\char0}% } \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~{\char0}% } \hbox{\amgr ~~~~~~~~{\char5}-----------------------{\char4}~~~~~~~~~~~~~~~{\char5}~{\char17}---------------------{\char4}% } \hbox{\amgr }} \vskip 0.35 truecm \par\noindent \centerline {\smbfb Fig.3.11} \endinsert \par By comparing times $t_1,\dots ,t_4$, we observe that the growth of an internal protuberance is always faster than the growth of an external one, indeed $$2J-(h+\l)<2J-(h-\l)\;\Rightarrow\;t_{1}M^{*}$: the internal component is subcritical, but not the external one, $R$ is supercritical and the system starting from $R$ will reach $\zero$; \item{} $L>L^{*}\;{\rm and}\;M>M^{*}$: both internal and external component are supercritical; $R$ is supercritical and the system starting from $R$ will reach $\piuuno$ by passing through $C(M_1-2,M_2-2)$ (internal growth is faster than external one); \item{} $L>L^{*}\;{\rm and}\;M~{\char0}~{\char0}~~~~~~~~~~~~~~~~{\char0}~{\char5}% -{\char3}~->~{\char0}~{\char0}~~~~~~~~~~~~~~~~{\char0}~~~~{\char0}} \hbox{\amgr ~~{\char0}~{\char0}~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~{\char0}~{\char2}% -{\char4}~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~{\char0}~~~~{\char0}} \hbox{\amgr ~~{\char0}~{\char0}~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~{\char0}~{\char0}% ~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~{\char0}~~~~{\char0}} \hbox{\amgr ~~{\char0}~{\char0}~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~{\char0}~{\char0}% ~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~{\char0}~~~~{\char0}} \hbox{\amgr ~~{\char0}~{\char5}----------------{\char4}~{\char0}~~~~{\char0}~{\char5}----------------{\char4}~{\char0}% ~~~~~~{\char0}~{\char5}----------------{\char4}~~~~{\char0}~} \hbox{\amgr ~~{\char5}--------------------{\char4}~~~~{\char5}--------------------{\char4}~~~~~~{\char5}-----------% ------------{\char4}} \hbox{\amgr ~~~~~~~~~~~$C(l,l)$~~~~~~~~~~~~~~~~~~~~$G(l,l)$~~~~~~~~~~~~~~~~~~~~~$R(l+1,l)$~~~~~~~~}} \vskip 0.35 truecm \par\noindent \centerline {\smbfb Fig.3.13} \endinsert \par\noindent As a consequence of the fact that it is always $t_10$, we have: $$l\Delta H_{22}\;\;\forall (i,j){\not\in}\{ (5,1),(3,2),(2,2)\}$. Hence, all the paths whose first step is different from (5,1) and (3,2) lead to a boundary configuration whose energy is greater than $H(C_{2,2})$. \par Starting from $C_{3,2}$ or $C_{5,1}$ an uphill path can continue by following one of the ways shown in Fig.4.2 and in Fig.4.3. It can be easily shown that the steps $(8,j)$ can be neglected as well. \par \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~{\char2}----------------------{\char3}} \hbox{\amgr ~~~~~~~~{\char0}~{\char2}--------------------{\char14}-{\char3}~} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~~~~{\char15}-{\char16}} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~$\left\{ \eqalign { +&\;\;\;\;\Delta H_{81}=6J{\char45}(h+\lambda)\cr {\char45}&\;\;\;\;\Delta H_{82}=2J+(h{\char45}\lambda)\cr}\right.$} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char5}--------------------{\char4}~{\char0}} \hbox{\amgr ~~~~~~~~{\char5}------------------------{\char4}}} \vskip 0.5 truecm \par\noindent \centerline {\smbfb Fig.4.3} \endinsert \par In conclusion, only the paths made by steps (3,2) and (5,1) can lead to a configuration whose energy is lower than $H(C_{2,2})$. \par Now, let $\s$ be an acceptable configuration such that the following conditions are satisfied: there exists just one cluster $c^0$ of $0$ spins which touches the sea of minuses namely the cluster $c^-$ winding around the torus, no minus spins are inside $c^0$; all plus spins are in a unique cluster $c^+$ included in $c^0$ and ${\hat\g^+}={\check\g^0}$. If $\s\in B$ then the following propositions are true: \itemitem{$i)$} $R(c^0)\equiv$ the external rectangle $(l_1+2)\times (l_2+2)$ of the frame $C$; \itemitem{$ii)$} $R(c^+)\equiv$ the internal rectangle $l_1\times l_2$ of the frame $C$; \itemitem{$iii)$} the intersection of each one of the four sides of $R(c^0)$ with $\hat\g^0$ contains at least a segment of length greater or equal to 2; \itemitem{$iv$)}the intersection of each one of the four sides of $R(c^+)$ with $\hat\g^+$ contains at least a segment of length greater or equal to 2. \par\noindent We prove $(i)$ by absurd: let us suppose that $R(c^0)$ is different from $(l_1+2)\times (l_2+2)$ and that ${\hat\g^+}$ is a rectangle. We can construct a downhill path which leads to a local minimum different from $C$ by filling with $0$ spins the region $R(c^0)\setminus c^+$. Thus $\s\not\in B$, and this is an absurd. $(ii)$ can be proved in a similar way. $(iii)$ is proved by absurd as well: suppose that the intersection between $\hat\g^0$ and one of the sides of $R(c^0)$ contains only isolated intervals of length 1, namely there is a certain number of spins $0$ with three minus spins among their nearest neighbour sites. By changing this $0$ spins into $-1$ we construct a configuration at a lower energy level and characterized by a cluster of $0$ spins $c'^0$ such that $R(c'^0)$ is different from the rectangle $(l_1+2)\times (l_2+2)$; then there exists a downhill path which connects $\s$ to a local minimum different from $C$. Hence the absurd $\s\not\in B$ is obtained. $(iv)$ is proved in a similar way. \par But, as we noticed before, all the uphill paths starting from $C$ and leading to configurations in $\partial B$ with energy smaller than $H(C_{2,2})$ necessarily can only be made by steps $(5,1)$ and $(3,2)$. \par It is clear that, by virtue of the necessary conditions stated above, we cannot reach $\partial B$ starting from $C$ with less than $l-1$ steps (5.1). On the other hand, since $S_1\in\partial B$, with more that $l-1$ steps $(5.1)$ we certainly get an energy larger that $H(S_1)$ and so a configuration which cannot be of minimal energy in $\partial B$. \par In this way we can only reach configurations with a unique cluster of pluses, so any boundary configuration with minimal energy is characterized by an external cluster $c^0$, such that the intersection between ${\hat\g^0}$ and all the sides of $R(c^0)$ is at least of length 2, and an internal cluster $c^+$, such that the intersection between ${\hat\g^+}$ and one of the sides of $R(c^+)$ has length 1 (see Fig.4.4). Among all these configurations it is easily seen that the one with lowest energy is $S_1$. \par \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~{\char2}---------{\char18}---{\char18}----------------{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~{\char0}~~~{\char0}~~~~~~~~~~~~~{\char45}~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~{\char45}~~{\char2}---{\char4}~~~{\char5}------------{\char3}~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~{\char0}~~~~~{\char2}-----{\char3}~~~~0~~~{\char0}~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~{\char15}-----{\char4}~~0~~{\char0}~~+~~{\char0}~~~~~~~~{\char0}~{\char45}~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~{\char2}---{\char4}~~~~~{\char5}----{\char3}~~~{\char5}---{\char16}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~{\char15}---{\char3}~~~{\char5}-----{\char3}~~~~~~~~{\char0}~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~{\char0}~~~~~~~~~{\char0}~~~+~~~~{\char0}~~~{\char2}---{\char16}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~{\char5}-----{\char3}~~~{\char5}--------{\char4}~0~{\char0}~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~{\char0}~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~{\char45}~~~~~{\char0}~0~{\char2}------------{\char4}~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~{\char5}---------{\char17}---{\char17}----------------{\char4}}} \vskip 0.5 truecm \par\noindent \centerline {\smbfb Fig.4.4} \endinsert \par In conclusion we have to compare $H(S_1)$ with $H(C_{2,2})$. Equality $\equ (4.3)$ follows from $l<{\widetilde L}$, $H(S_1)-H(C)= (h+\l)(l-1)$ and $H(C_{2,2})-H(C)=2J-(h-\l)$. \par Now we want to apply to the description of the first escape from $B$ the approach developed in [OS1], which is based on the properties of the above defined sets called cycles. \par It is easy to see that the basin of attraction $B:=B(C(l_1,l_2))$ defined in $\equ (2.6b)$ satisfies the following properties: \itemitem{$i)$} $B$ is connected; \itemitem{$ii)$} $S_1\subset\partial B$, and $$\min_{\s\in\partial B} H(\s)=H(S_1), \; \min_{\s\in\partial B\setminus S_1} H(\s)>H(S_1)$$ \itemitem{$iii)$} $\forall \h\in S_1$ there exists a path $\o :\h\rightarrow C$ such that $\forall\s\in\o\setminus \{\h\}$ one has $\s\in B$ and $H(\s)From Proposition 3.7 in [OS1], from reversibility of the dynamics (see Lemma 1 in [KO1]) and from $\equ (4.3.3)$ we easily get that $\forall\s\in {\bar B}$ $$\lim_{\b\to\infty} P_{\s}(\s_{\t_{(B\cup\partial B)^c}-1}\in S_1)=1\;\; .\Eq (4.3.4)$$ \par Since $H(S_1)-H(C)=(h+\l)(l-1)$ we deduce that for every $\e>0$ $$\lim_{\b\to\infty} P_C( e^{\b (h+\l)(l-1)-\b\e}<\t_{\partial B}~{\char0}~{\char0}~~~~~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~{\char0}~{\char0}~~~~~~~~~~~~~{\char0}~~~{\char0}~~~~~{\char0}~{\char0}~~~~~~~~~~~~~{\char0}~~~{\char0}% ~~~~~{\char0}~{\char0}~~~~~~~~~~~~~{\char2}-{\char4}~{\char0}} \hbox{\amgr ~~~~~{\char0}~{\char0}~~~~~~~~~~~~~{\char0}~~~{\char0}~~~~~{\char0}~{\char0}~~~~~~~~~~~~~{\char0}~~~{\char0}% ~~~~~{\char0}~{\char0}~~~~~~~~~~~~~{\char0}~~~{\char0}} \hbox{\amgr ~~~~~{\char0}~{\char5}-------------{\char4}~~~{\char0}~~~~~{\char0}~{\char5}-------------{\char4}~~~{\char0}% ~~~~~{\char0}~{\char5}-------------{\char4}~~~{\char0}} \hbox{\amgr ~~~~~{\char5}-------------------{\char4}~~~~~{\char5}-------------------{\char4}~~~~~{\char5}-------------% ------{\char4}}} \vskip 0.5 truecm \par\noindent \centerline {\smbfb Fig.4.5} \endinsert \par In Appendix 1 we give a general argument showing that, with high probability, our process, possibly after many attempts, soon or later, will eventually get out of $B \cup \partial B$ through $S_1$ reaching $R_1$ before touching any other local minimum and: $$\eqalign { \lim_{\beta\to\infty}P_{C}&(\tau_{R_{1}}<\tau_{\piuuno})=1\cr \lim_{\beta\to\infty}P_{C}&(\tau_{R_{1}}0$, $e^{\beta[\min_{\sigma\in\partial B_{1}}H(\sigma)-H(R_{1})]+\beta\e}$ is an upper bound, in the limit $\b\to\infty$ to the first hitting time to $R_2$ of the Markov chain starting from $R_1$. \par In conclusion we can say that the Markov chain starting from $C$ visits smaller and smaller local minima untill it reaches the configuration $\menouno$; this completes the proof of the statement $P_C(\t_{\menouno}<\t_{\piuuno}) {\buildrel\beta\rightarrow\infty\over\longrightarrow}\; 1$. \par Each step of the shrinking process is characterized by a typical time $t_{\beta}$ whose asymptotic behaviour, exponentially in $\beta$, is known in the sense that we control $$\lim_{\beta\to\infty} {1\over \beta} \log t_{\beta}\;\; ;$$ we say that: $$t^1_{\beta}, t^2_{\beta}\; {\rm are} \; logarithmically\; equivalent\; \Leftrightarrow \; \lim_{\b\to\infty} {1\over\b} t^1_{\beta}= \lim_{\b\to\infty} {1\over\b} t^2_{\beta}\;\; .$$ \par By using Markov property we can say that the typical time of the whole shrinking event is given by the largest time among all the {\it partial shrinking times}. Then the proof of Proposition 4.1 is completed in the case $l<{\widetilde L}$ when $C$ is a rectangular frame. \par Next, we consider the case when $C$ is a squared frame: the boundary configuration $S_3$ is now the one represented in Fig.4.8, $H(S_3)-H(R_1)=(h+\l)(l-2)$ and $\min_{\s\in\partial B_1} H(\s) =H({\cal S}_3)$ if $l<\left[ {3\over 2}{h\over\l}+{1\over 2}\right] +1$. We obtain results similar to those obtained in the previous case of a rectangular frame. \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char2}---------------{\char3}~~} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char2}-{\char3}~~~~~~~~~~~{\char0}~~} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~{\char5}-------{\char3}~~~{\char0}~~~~} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~{\char0}~~~{\char0}~~} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~{\char0}~~~{\char0}~~$l+2$} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~{\char0}~~~{\char0}~~} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~{\char0}~~~{\char0}~} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char5}---------{\char4}~~~{\char0}~~} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char5}---------------{\char4}~~} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$l+2$} \hbox{\amgr }} \vskip 0.5 truecm \par\noindent \centerline {\smbfb Fig.4.8} \endinsert \par Now we consider the frame $C:=C(l_1,l_2)$; we suppose that ${\widetilde L}\leq l:=\min \{l_1,l_2\}H(S_4) \;\; ,\Eq (4.17)$$ namely $$U(B)=S_4\;\; ;\Eq (4.17.1)$$ we remark that $H(S_4)-H(C)=2J-(h-\l)$. \par \bigskip \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~$\phantom {S_{4,\perp}}$~~~~~{\char2}---------------------{\char3}~~~~~~~~~{\char2}---------------------{\char3}} \hbox{\amgr ~~~~~~~~$\phantom {S_{4,\perp}}$~~~~~{\char0}~{\char2}-----------------{\char3}~{\char0}~~~~~~~~~{\char0}~{\char2}-----------------{\char3}~{\char0}} \hbox{\amgr ~~~~~~~~$\phantom {S_{4,\perp}}$~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char5}-{\char3}~~~~~{\char2}-{\char4}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~$ {S_{4,\perp}}$~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char2}-{\char4}~~~~~{\char5}-{\char3}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}~$l+2$} \hbox{\amgr ~~~~~~~~$\phantom {S_{4,\perp}}$~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~$\phantom {S_{4,\perp}}$~~~~~{\char0}~{\char5}-----------------{\char4}~{\char0}~~~~~~~~~{\char0}~{\char5}-----------------{\char4}~{\char0}} \hbox{\amgr ~~~~~~~~$\phantom {S_{4,\perp}}$~~~~~{\char5}---------------------{\char4}~~~~~~~~~{\char5}---------------------{\char4}} \hbox{\amgr ~~~~~~~~$\phantom {S_{4,\perp}}$~~~~~~~~~~~~~$m+2$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$m+2$~~~~~~~~~~~~} \hbox{\amgr } \hbox{\amgr } \hbox{\amgr ~~~~~~~~$\phantom {S_{4,\parallel}}$~~~~~~~~~~~~~~{\char2}-{\char3}} \hbox{\amgr ~~~~~~~~$\phantom {S_{4,\parallel}}$~~~~~{\char2}--------{\char4}~{\char5}----------{\char3}~~~~~~~~~{\char2}---------------------{\char3}} \hbox{\amgr ~~~~~~~~$\phantom {S_{4,\parallel}}$~~~~~{\char0}~{\char2}-----------------{\char3}~{\char0}~~~~~~~~~{\char0}~{\char2}-----------------{\char3}~{\char0}} \hbox{\amgr ~~~~~~~~$\phantom {S_{4,\parallel}}$~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~$ {S_{4,\parallel}}$~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}~$l+2$} \hbox{\amgr ~~~~~~~~$\phantom {S_{4,\parallel}}$~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~$\phantom {S_{4,\parallel}}$~~~~~{\char0}~{\char5}-----------------{\char4}~{\char0}~~~~~~~~~{\char0}~{\char5}-----------------{\char4}~{\char0}} \hbox{\amgr ~~~~~~~~$\phantom {S_{4,\parallel}}$~~~~~{\char5}---------------------{\char4}~~~~~~~~~{\char5}--------{\char3}~{\char2}----------{\char4}} \hbox{\amgr ~~~~~~~~$\phantom {S_{4,\parallel}}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char5}-{\char4}~~} \hbox{\amgr ~~~~~~~~$\phantom {S_{4,\parallel}}$~~~~~~~~~~~~~$m+2$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$m+2$~~~~~~~~~~~~}} \vskip 0.5 truecm \par\noindent \centerline {\smbfb Fig.4.9} \endinsert \par Without loss of generality we suppose that $l=l_2$ and $m=l_1$. By arguments similar to those used before it can be proved that in a typical time $e^{[2J-(h-\l)]}$ the Markov chain starting from $C$, with high probability, will visit $R_{2,\perp}:=R(l_1,l_2;l_1+3,l_2+2)$ or $R_{2,\parallel}:=R(l_1,l_2;l_1+2,l_2+3)$. The symbol $\perp$ denotes the fact that the frame is growing in a direction perpendicular to its shortest side (see Fig.4.10). \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~$\phantom {R_{2,\perp}}$~~~~~{\char2}-----------------------{\char3}~~~~~{\char2}-----------------------{\char3}} \hbox{\amgr ~~~~~~~~$\phantom {R_{2,\perp}}$~~~~~{\char0}~{\char2}-----------------{\char3}~~~{\char0}~~~~~{\char0}~~~{\char2}-----------------{\char3}~{\char0}} \hbox{\amgr ~~~~~~~~$\phantom {R_{2,\perp}}$~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~~~{\char0}~~~~~{\char0}~~~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~$ {R_{2,\perp}}$~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~~~{\char0}~~~~~{\char0}~~~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}~$l+2$} \hbox{\amgr ~~~~~~~~$\phantom {R_{2,\perp}}$~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~~~{\char0}~~~~~{\char0}~~~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~$\phantom {R_{2,\perp}}$~~~~~{\char0}~{\char5}-----------------{\char4}~~~{\char0}~~~~~{\char0}~~~{\char5}-----------------{\char4}~{\char0}} \hbox{\amgr ~~~~~~~~$\phantom {R_{2,\perp}}$~~~~~{\char5}-----------------------{\char4}~~~~~{\char5}-----------------------{\char4}} \hbox{\amgr ~~~~~~~~$\phantom {R_{2,\perp}}$~~~~~~~~~~~~~$m+3$~~~~~~~~~~~~~~~~~~~~~~~~~~$m+3$~~~~~~~~~~~~~~~} \hbox{\amgr } \hbox{\amgr } \hbox{\amgr ~~~~~~~~$\phantom {R_{2,\parallel}}$~~~~~{\char2}---------------------{\char3}~~~~~~~~~} \hbox{\amgr ~~~~~~~~$\phantom {R_{2,\parallel}}$~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~{\char2}---------------------{\char3}} \hbox{\amgr ~~~~~~~~$\phantom {R_{2,\parallel}}$~~~~~{\char0}~{\char2}-----------------{\char3}~{\char0}~~~~~~~~~{\char0}~{\char2}-----------------{\char3}~{\char0}} \hbox{\amgr ~~~~~~~~$\phantom {R_{2,\parallel}}$~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~$ {R_{2,\parallel}}$~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}~$l+3$} \hbox{\amgr ~~~~~~~~$\phantom {R_{2,\parallel}}$~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~$\phantom {R_{2,\parallel}}$~~~~~{\char0}~{\char5}-----------------{\char4}~{\char0}~~~~~~~~~{\char0}~{\char5}-----------------{\char4}~{\char0}} \hbox{\amgr ~~~~~~~~$\phantom {R_{2,\parallel}}$~~~~~{\char5}---------------------{\char4}~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~$\phantom {R_{2,\parallel}}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char5}---------------------{\char4}} \hbox{\amgr ~~~~~~~~$\phantom {R_{2,\parallel}}$~~~~~~~~~~~~~$m+2$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$m+2$~~~~~~~~~~~~}} \vskip 0.5 truecm \par\noindent \centerline {\smbfb Fig.4.10} \endinsert \par We denote by $B_{2,\perp}$ and $B_{2,\parallel}$ the basins of attraction of $R_{2,\perp}$ and $R_{2,\parallel}$; their boundaries are respectively denoted by $\partial B_{2,\perp}$ and $\partial B_{2,\parallel}$. By considering all the uphill paths starting from $R_{2,\perp}$ and $R_{2,\parallel}$ we get that $$\eqalign { \min_{\sigma\in\partial B_{2,\perp}}H(\sigma)=H(S_{4})\cr \min_{\sigma\in\partial B_{2,\parallel}}H(\sigma)=H(S_{4})\cr} \;\; ,\Eq (4.18)$$ more precisely $$U(B_{2,\perp})=S_4,\;\; U(B_{2,\parallel})=S_4 \;\; .\Eq (4.18.1)$$ Indeed the most relevant inequalities in the proof of $\equ (4.18)$ are the following ones $$\eqalign{ (h+\lambda)(l-1)>&2J-(h-\lambda)>2J-(h+\lambda)>(h-\lambda)(l+1)\cr (h+\lambda)(l-1)>&2J-(h-\lambda)>2J-(h+\lambda)>(h-\lambda)(m+1)\cr}\;\; . \Eq (4.19)$$ We remark that $H(S_{4})-H(R_{2,\perp})=(h-\lambda)(l+1)$ and $H(S_{4})-H(R_{2,\parallel})=(h-\lambda)(m+1)$. In order to prove the first one of the equalities $\equ (4.19)$ we notice that $$\eqalign { l\geq {\widetilde L}&\Rightarrow (h+\lambda)(l-1)>2J-(h-\lambda)\cr l+2< {\widetilde M}&\Rightarrow 2J-(h+\lambda)>(h-\lambda)(l+1)\cr}\;\; .$$ In order to prove the second one we notice that $$l\geq {\widetilde L}\Rightarrow m^{*}(l)+2\leq {\widetilde M}$$ and that $$m< m^{*}(l)\Rightarrow m+2<{\widetilde M}\Rightarrow (h-\lambda)(m+1)<2J-(h+\lambda)\;\; .$$ \par Starting from $R_{2,\perp}$ or from $R_{2,\parallel}$ the system will typically go back to $C$ before visiting other frames; these phenomena take place, respectively, in the two typical times $e^{\beta(h-\lambda)(l+1)}$ and $e^{\beta(h-\lambda)(m+1)}$. It appears clear that the system, before eventually leaving $C$ to reach another frame, will wander, performing random oscillations, in the union of the basins $B$, $B_{2,\perp}$ and $B_{2,\parallel}$. Then, in order to understand whether the frame will shrink or grow we have to describe its behaviour in a larger basin, containing $B\cup B_{2,\perp} \cup B_{2,\parallel}$. This basin is denoted by ${\cal D}$ and it is defined as follows $${\cal D}:= \{\h :{\rm every\; downhill\; path\; starting\; from\;} \h\; {\rm ends\; in\;} C\; {\rm or}\; R_{2,\perp}\; {\rm or}\; R_{2,\parallel} \}\;\; .\Eq (4.20)$$ \par We denote by $S_{5,\perp}$ and $S_{5,\parallel}$ the configurations obtained by attaching a unit square protuberance to the free side of the internal rectangle of $R_{2,\perp}$ and $R_{2,\parallel}$ (see Fig.4.11). By considering all the uphill paths starting from $C$, $R_{2,\perp}$ and $R_{2,\parallel}$, we are able to examine all the configurations in $\partial {\cal D}$. We get: $$\min_{\sigma\in\partial {\cal D}}H(\sigma)=H(S_{1}),\;\;\;\; U({\cal D})=S_1\;\; .\Eq (4.21)$$ The most relevant inequalities in the proof of equation $\equ (4.21)$ are $$\eqalign{ lH(S_1)$ and $H(S_{5,\parallel})>H(S_1)$. Of course it is always $H(S_{5,\parallel})From ${\cal D}$ one can easily obtain, by suitably cutting in energy, a cycle having the same minimal saddles in its boundary:\par take the maximal connected set $\bar {\cal D}$ of configurations containing $ C $ with energy less than $H(S_1)$. Since it is easy to see that properties $i)$, $ii)$, $iii)$ of $B=B(C(l_1,l_2))$ are still verified with ${\cal D}$ in place of $B$ for $\widetilde L \leq l < l^*, \;\; m < m^* (l)$ one immediately gets: ${\cal S}(\bar {\cal D}) \ni S_1$; moreover, $\forall\s\in {\cal D}$, $$\lim_{\b\to\infty} P_{\s}(\s_{\t_{({\cal D}\cup\partial {\cal D})^c}-1}\in S_1)=1 \;\; $$ and for every $\e>0$ $$\lim_{\b\to\infty} P_C( e^{\b (h+\l)(l-1)-\b\e}<\t_{\partial {\cal D}} H(S_{5,\parallel})$, $H(S_4)< H(S_{5,\parallel})$. With the usual arguments one can prove that $$\min_{\s\in\partial {\cal D}} H(\s)=H(S_{5,\parallel}),\;\;\; U({\cal D})=S_{5,\parallel}\;\; ;\Eq (4.23)$$ hence the frame $C$ is supercritical and the system starting from $C$ will hit ${\piuuno}$ in a typical time $e^{\beta\{[2J-(h-\lambda)]-(h-\lambda)(m+1)+[2J-(h+\lambda)]\}}$. \par In the case ${\widetilde L}\leq lH(S_4)$. Hence the frame is supercritical and the typical escape time is $e^{\beta\{[2J-(h-\lambda)]-(h-\lambda)(m+1)+[2J-(h+\lambda)]\}}$. In this case the most important inequalities are $$\eqalign { (h+\lambda)(l-1)>&[2J-(h-\lambda)]-(h-\lambda)(l+1)+[2J-(h+\lambda)]>\cr >&[2J-(h-\lambda)]-(h-\lambda)(m+1)+[2J-(h+\lambda)]\cr} \;\; ,\Eq (4.24)$$ we remark that $H(S_1)-H(C)=(h+\lambda)(l-1)$, $H(S_{5,\perp})-H(C)=[2J-(h-\lambda)]-(h-\lambda)(l+1) +[2J-(h+\lambda)]$ and $H(S_{5,\parallel})-H(C)= [2J-(h-\lambda)]-(h-\lambda)(m+1)+[2J-(h+\lambda)]$. \par In the case $l^*\leq l<{\widetilde M}-2$ and ${\widetilde M}-2\leq m$ we have that $\min_{\s\in\partial {\cal D}} H(\s)=H(S_{5,\parallel})$ and $H(S_{5,\parallel})~~{\char0}~~~~{\char0}~~~~~~~~~~~{\char0}% ~~~~~~{\char0}~${\hat L}+2$} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~{\char0}~~~~~~~~~~~{\char0}~~~~~~{\char0}~~~~~~~{\char0}~~~~{\char0}~~~~~~~~~~~{\char0}% ~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~{\char0}~~~~~~~~~~~{\char0}~~~~~~{\char0}~~~~~~~{\char0}~~~~{\char0}~~~~~~~~~~~{\char0}% ~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~{\char0}~~~~~~~~~~~{\char0}~~~~~~{\char0}~~~~~~~{\char0}~~~~{\char0}~~~~~~~~~~~{\char0}% ~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~{\char0}~~~~~~~~~~~{\char0}~~~~~~{\char0}~~~~~~~{\char0}~~~~{\char0}~~~~~~~~~~~{\char0}% ~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~{\char0}~~~~~~~~~~~{\char0}~~~~~~{\char0}~~~~~~~{\char0}~~~~{\char0}~~~~~~~~~~~{\char0}% ~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~{\char5}-----------{\char4}~~~~~~{\char0}~~~~~~~{\char0}~~~~{\char5}-----------{\char4}% ~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~{\char5}-----------------------{\char4}} \hbox{\amgr ~~~~~~~~~~~~~~{\char5}-----------------------{\char4}~~~~~~~} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~$M$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$M$~~~~~~~~~~~}} \vskip 0.5 truecm \par\noindent \centerline {\smbfb Fig.4.15} \endinsert \par Then, we can conclude that in the cases $(2)$ and $(3)$ the frame $C(L_1,L_2)$ is eventually reached, but $C(L_1,L_2)$ is subcritical, hence $R$ is subcritcal, as well. For similar reasons in the cases $(8)$ and $(9)$ the birectangle $R$ is supercritical. \par The typical shrinking time is given by $$\eqalign{ \max\{e^{\b (M-1)(h-\l)},e^{\b (L-1)(h+\l)}\}& \; {\rm if}\; {\hat L}+2\leq M\cr \max\{e^{\b ({\hat L}+1)(h-\l)},e^{\b (L-1)(h+\l)}\}& \; {\rm if}\; {\hat L}+2> M\cr}\;\; .$$ \par With similar arguments it can be shown that in cases $(4)$ and $(5)$ the system, starting from $R$, hits $C(M-2,{\hat L})$ in a typical time $e^{2J-(h+\l)}$. Hence, the birectangle $R$ is subcritical and the typical shrinking time is $e^{2J-(h+\l)}$. In the cases $(10)$ and $(11)$ the birectangle $R$ is supercritical, as a consequence of the supercriticality of the frame $C(M-2,{\hat L})$. \par With arguments similar to those used before it can also be seen that in the case $(6)$ the birectangle is supercritical since it first evolves towards the frame $C (M - 2, \hat M - 2)$ which is a supercritical frame since $ \hat M - 2 > l^* $ with our choice of the paremeters.\par Finally, in the case $(7)$, the birectangle is easily seen to be supercritical. Indeed it follows from an argument similar to the corresponding one valid for the standard Ising model that starting from a configuration with $M\ge M^*$, we get $\zero$ before $\piuuno$ in a time of order $e^{\beta [2J-(h-\l)]}$ with high probability for large $\beta$. Then, starting from $\zero$ we tipically follow an Ising--like nucleation path (see [NS1], [S1]) leading to $\piuuno$ through the saddles ${\cal S}(\zero ,\piuuno )$. These saddles are given by configurations with precisely one cluster of pluses (in the sea of zeroes), this cluster is given by a rectangle $L^*\times (L^*-1)$ with a unit square protuberance attached to one of its longest sides. It is immediate to verify that $$H({\cal S}(\zero , \piuuno)) < H({\cal P})\;\; .$$ \par The proof of Proposition 4.2 is complete. $\square$ \par \bigskip We consider, now, a plurirectangle $R$. We denote by $M_1$ and $M_2$ the lengths of the sides of the external rectangle, by $L_{1,i}$ and $L_{2,i}$ $\forall i=1,...,k^+$ the lengths of the sides of the $k^+$ internal rectangles $R^+_i$ and we define $M:=\min \{M_1,M_2\}$ and $L_i:=\min \{L_{1,i},L_{2,i}\}\; \forall i=1,...,k^+$. In order to state conditions of subcriticality and supercriticality for such configurations, we must introduce the rectangle $R^+$ defined as the rectangular envelpe of the union of all the internal supercritical rectangles. We denote by $L_{1,R^+}$ and $L_{2,R^+}$ the lenghts of its sides and we define $L_{R^+}:=\min \{L_{1,R^+},L_{2,R^+}\}$ and ${\hat L}_{R^+}:=\max \{L_{1,R^+},L_{2,R^+}\}$. Suppose that $\exists i\in\{1,2,...,k^+\}$ such that $L_i\geq L^*$, we denote by ${\bar R}$ the birectangle obtained by removing all the internal rectangles and by filling up with plus spin the rectangle $R^+$. Finally we state the following proposition %%%%%%%%%%%%%%%%%Proposition 4.3 \vskip 0.35 truecm \noindent {\bf Proposition 4.3.}\par\noindent If one of the two following conditiones is satisfied \itemitem{1)} $L_i M-2$. During the second phase of the contraction the system reaches a configuration characterized by an external rectangle whose sides are $M$ and ${\hat L}^{(1)}+2$. The ``free" side of the external rectangle is eventually ${\hat L}^{(1)}+2$. If $({\hat L}^{(1)}+1)(h-\lambda)<(L_{i}-1)(h+\lambda)\; \forall i\in I^{(1)}$ the external rectangle shrinks in a direction perpendicular to its ``free" side untill it reaches $R^{(1)}$; and then the shrinkig goes on as we have described before. If there exists an internal rectangle $R^+_i$ such that $({\hat L}^{(1)}+1)(h-\lambda)>(L_{i}-1)(h+\lambda)$ it disappers before anything else can happen. Then the contraction goes on as described before. In conclusion we have proved that in the case $(1)$ the plurirectangle $R$ is subcritical. \par In the case $(2)$ the proof of Proposition 4.3 can be achieved with arguments similar to those used in the case $(1)$. $\square$ \par \bigskip \vfill\eject \numsec=5\numfor=1 {\bf Section 5. Comparison between special saddles.} \par Let us consider a subcritical frame or birectangle; we say that such a configuration is {\it almost--supercritical} iff it can be transformed into a supercritical minimum by attaching to one of its internal or external sides a whole slice. By attaching a slice to an internal or external side of a birectangle (or, in particular, of a frame) we mean transforming from $-1$ to $0$ the value of the spins in the row or column adjacent externally to this side. ``Removing a slice" is the inverse operation of ``attaching a slice". \par Let us consider, now, a supercritical frame or birectangle; we say that such a configuration is {\it just--supercritical} iff it can be transformed into a subcritical minimum by removing a whole slice from one of its internal or external sides. \par Let us consider an almost supercritical frame or birectangle, we denote by $u$ the internal or external side such that by attaching to it a whole slice we obtain a supercritical configuration. We call {\it special saddle} the configuration obtained by attaching to $u$ a plus unit protuberance, if $u$ is an internal side, or a zero unit protuberance, if $u$ is an external one. \par \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~$\phantom {{\cal P}_1={\cal P}_{1,a}\;\; {\rm if}\;{\it \delta } <{h+{\it \lambda }\over 2h}}$~{\char2}----------------{\char3}~~$\phantom {l^*+2}$~~{\char2}--------------{\char3}~~~~~~~~~% ~~~~~~~} \hbox{\amgr ~$\phantom {{\cal P}_1={\cal P}_{1,a}\;\; {\rm if}\;{\it \delta } <{h+{\it \lambda }\over 2h}}$~{\char0}~{\char2}----------{\char3}~~~{\char0}~~$\phantom {l^*+2}$~~{\char0}~~~~~{\char2}-{\char3}% ~~~~~~{\char0}} \hbox{\amgr ~$\phantom {{\cal P}_1={\cal P}_{1,a}\;\; {\rm if}\;{\it \delta } <{h+{\it \lambda }\over 2h}}$~{\char0}~{\char0}~~~~~~~~~~{\char0}~~~{\char0}~~$\phantom {l^*+2}$~~{\char0}~{\char2}---{\char4}% ~{\char5}----{\char3}~{\char0}} \hbox{\amgr ~$ {\cal P}_1={\cal P}_{1,a}\;\; {\rm if}\;{\it \delta } <{h+{\it \lambda }\over 2h}$~{\char0}~{\char0}~~~~~~~~~~{\char0}~~~{\char0}~~$\phantom {l^*+2}$~~{\char0}~{\char0}~~~~~~~~% ~~{\char0}~{\char0}~${\cal P}_1={\cal P}_{1,b}\;\; {\rm if}\;{\it \delta }>{h+{\it \lambda }\over 2h}$} \hbox{\amgr ~$\phantom {{\cal P}_1={\cal P}_{1,a}\;\; {\rm if}\;{\it \delta } <{h+{\it \lambda }\over 2h}}$~{\char0}~{\char0}~~~~~~~~~~{\char5}-{\char3}~{\char0}~~$\phantom {l^*+2}$~~{\char0}~{\char0}~% ~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~$\phantom {{\cal P}_1={\cal P}_{1,a}\;\; {\rm if}\;{\it \delta } <{h+{\it \lambda }\over 2h}}$~{\char0}~{\char0}~~~~~~~~~~{\char2}-{\char4}~{\char0}~~${l^*+2}$~~{\char0}~{\char0}~% ~~~~~~~~~{\char0}~{\char0}~~} \hbox{\amgr ~$\phantom {{\cal P}_1={\cal P}_{1,a}\;\; {\rm if}\;{\it \delta } <{h+{\it \lambda }\over 2h}}$~{\char0}~{\char0}~~~~~~~~~~{\char0}~~~{\char0}~~$\phantom {l^*+2}$~~{\char0}~{\char0}~~~~~~~~% ~~{\char0}~{\char0}} \hbox{\amgr ~$\phantom {{\cal P}_1={\cal P}_{1,a}\;\; {\rm if}\;{\it \delta } <{h+{\it \lambda }\over 2h}}$~{\char0}~{\char0}~~~~~~~~~~{\char0}~~~{\char0}~~$\phantom {l^*+2}$~~{\char0}~{\char0}~~~~~~~~% ~~{\char0}~{\char0}} \hbox{\amgr ~$\phantom {{\cal P}_1={\cal P}_{1,a}\;\; {\rm if}\;{\it \delta } <{h+{\it \lambda }\over 2h}}$~{\char0}~{\char5}----------{\char4}~~~{\char0}~~$\phantom {l^*+2}$~~{\char0}~{\char5}--------% --{\char4}~{\char0}} \hbox{\amgr ~$\phantom {{\cal P}_1={\cal P}_{1,a}\;\; {\rm if}\;{\it \delta } <{h+{\it \lambda }\over 2h}}$~{\char5}----------------{\char4}~~$\phantom {l^*+2}$~~{\char5}--------------{\char4}} \hbox{\amgr ~$\phantom {{\cal P}_1={\cal P}_{1,a}\;\; {\rm if}\;{\it \delta } <{h+{\it \lambda }\over 2h}}$~~~~~~${l^*+2}$~~~~~~~~~~~$\phantom {l^*+2}$~~~~~~~$l^*+1$} \hbox{\amgr } \hbox{\amgr } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal P}_2}$~~~~~~{\char2}--------------{\char3}~~~~~~~~~~~~~~~~} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal P}_2}$~~~~~~{\char0}~~~~~~~~~~~~~~{\char0}~~} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal P}_2}$~~~~~~{\char0}~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal P}_2}$~~~~~~{\char0}~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal P}_2}$~~~~~~{\char0}~~~~~~~~~~~~~~{\char5}-{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ {\cal P}_2$~~~~~~{\char0}~~~~~~~~~~~~~~{\char2}-{\char4}~~$M^*$} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal P}_2}$~~~~~~{\char0}~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal P}_2}$~~~~~~{\char0}~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal P}_2}$~~~~~~{\char0}~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal P}_2}$~~~~~~{\char5}--------------{\char4}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal P}_2}$~~~~~~~~~~~~$M^*{\char 45}1$~~~~~~~~~~~~~~~~~~~~~~~}} \vskip 0.5 truecm \par\noindent \centerline {\smbfb Fig.5.1} \par Let us consider the set ${\hat {\cal P}}:=({\cal P}_1\cup {\cal P}_2)\subset \O_{\L}$ with ${\cal P}_1$ and ${\cal P}_2$ the set of special saddles shown in Fig.5.1, where we have used the following definition $$\d:=l^*-{2J-(h-\l)\over h}\in\rbrack 0,1\lbrack\;\; .\Eq (5.1)$$ We state the following lemma: %%%%%%%%%%%%%%%%%Lemma \vskip 0.5 truecm \noindent {\bf Lemma 5.1.}\par\noindent For any special saddle $S\not\in {\hat {\cal P}}$ it there exists $S^*\in {\hat {\cal P}}$ such that $$H(S)>H(S^*)\;\; .$$ %%%%%%%%%%%%%%%%%%%End of Lemma. \par\noindent \vskip 0.5 truecm \par Before starting the proof, we observe that the frame $C(l^*,l^*)$ is supercritical and $C(l^*-1,l^*-1)$ is subcritical for any choice of the parameters $\l$ and $h$; indeed it can be proved that $$m^*(l^*-1)\ge l^*\;{\rm for\; any\; value\; of\;} h\; {\rm and}\; \l , \Eq (5.1.1)$$ (see $(5.5)$). On the other hand we remark that the criticality of the frame $C(l^*-1,l^*)$ depends on the value of the real number $\d$ defined in $\equ (5.1)$. By comparing the energies of the two configurations shown in Fig.5.2 one can easily convince himself that $$\eqalign{ C(l^*-1,l^*) &\;\; {\rm subcritical\; iff}\;\d< {h+\l\over 2h}\cr C(l^*-1,l^*) &\;\; {\rm supercritical\; iff}\;\d> {h+\l\over 2h}\cr }\;\; ,$$ we observe that ${h+\l\over 2h}\in\rbrack 0,1\lbrack$ if ${h\over\l}>1$. This explains the reason of the twofold definition of the configuration ${\cal P}_1$. \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~$\phantom {l^*+2}$~~{\char2}----------------{\char3}~~~~~~~~~~~~~~{\char2}------------------{\char3}~~~~~% ~~~~~~~} \hbox{\amgr ~~~~~~~~~~~$\phantom {l^*+2}$~~{\char0}~{\char2}-{\char3}~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~{\char0}~{\char2}--------% ----{\char3}~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~$\phantom {l^*+2}$~~{\char0}~{\char0}~{\char5}----------{\char3}~{\char0}~~~~~~~~~~~~~~{\char0}~{\char0}~% ~~~~~~~~~~~{\char0}~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~$\phantom {l^*+2}$~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~% ~~~~{\char0}~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~$\phantom {l^*+2}$~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~% ~~~~{\char5}-{\char3}~{\char0}} \hbox{\amgr ~~~~~~~~~~~$l^*+2$~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~% ~~~~{\char2}-{\char4}~{\char0}~~$l^*+2$} \hbox{\amgr ~~~~~~~~~~~$\phantom {l^*+2}$~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~% ~~~~{\char0}~~~{\char0}~~~~} \hbox{\amgr ~~~~~~~~~~~$\phantom {l^*+2}$~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~% ~~~~{\char0}~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~$\phantom {l^*+2}$~~{\char0}~{\char5}------------{\char4}~{\char0}~~~~~~~~~~~~~~{\char0}~{\char5}--------% ----{\char4}~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~$\phantom {l^*+2}$~~{\char5}----------------{\char4}~~~~~~~~~~~~~~{\char5}------------------{\char4}} \hbox{\amgr ~~~~$\phantom {l^*+2}$~~~~~~~~~~~~~~$l^*+1$~~~~~~~~~~~~~~~~~~~~~~~~~~$l^*+2$}} \vskip 0.5 truecm \par\noindent \centerline {\smbfb Fig.5.2} \par \vskip 0.5 truecm \par\noindent {\it Proof of lemma} 5.1. \vskip 0.35 truecm \par\noindent Let us suppose that $\d <{h+\l\over 2h}$. One can prove that for any $l$ such that ${\widetilde L}\le l\le l^*-1$ $$m^*(l)\ge l^*+1\;\; .\Eq (5.2)$$ First of all we observe that $m^*(l)$ is a decreasing function of $l$, more precisily one can easily prove that $$m^*(l-1)\ge m^*(l)+1\;\;\forall l\in [{\widetilde L},l^*-1]\;\; .\Eq (5.3)$$ Therefore in order to get a lower bound to $m^*(l)$ it is sufficient to evaluate $m^*(l^*-1)$; with some algebra one can easily obtain $$m^*(l^*-1)=l^*+\left[ (1-\d) {2h\over h-\l}\right]\;\; .\Eq (5.4)$$ Then, $$\d <{h+\l\over 2h}\Rightarrow (1-\d){2h\over h-\l}>1\Rightarrow m^*(l^*-1)\ge l^*+1\;\; ;$$ this completes the proof of inequality $\equ (5.2)$. We remark that the validity of the equations $\equ (5.3)$ and $\equ (5.4)$ does not depend on the value of the real number $\d$. \par Now, in order to prove Lemma 5.1 we have to examine all the possible special saddles. \vskip 0.35 truecm \par\noindent {\it Case C1}. \par We consider the special saddle $C_{1}(m)$ in Fig.5.3. \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~$\phantom {C_1(m)}$~~~~{\char2}------------------{\char3}~~~~~~~~~~~~} \hbox{\amgr ~~~~~~~$\phantom {C_1(m)}$~~~~{\char0}~{\char2}------------{\char3}~~~{\char0}} \hbox{\amgr ~~~~~~~$\phantom {C_1(m)}$~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~~~{\char0}} \hbox{\amgr ~~~~~~~$\phantom {C_1(m)}$~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~~~{\char0}~~$m+2$} \hbox{\amgr ~~~~~~~$\phantom {C_1(m)}$~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char5}-{\char3}~{\char0}} \hbox{\amgr ~~~~~~~$\phantom {C_1(m)}$~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char2}-{\char4}~{\char0}~~~} \hbox{\amgr ~~~~~~~$ {C_1(m)}$~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~~~{\char0}~~~~} \hbox{\amgr ~~~~~~~$\phantom {C_1(m)}$~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~~~{\char0}~~~~~~~~${\rm with}$~$l^*\le m\le m^*(l^*{\char45}1){\char45}1$} \hbox{\amgr ~~~~~~~$\phantom {C_1(m)}$~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~~~{\char0}~~~~} \hbox{\amgr ~~~~~~~$\phantom {C_1(m)}$~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~~~{\char0}} \hbox{\amgr ~~~~~~~$\phantom {C_1(m)}$~~~~{\char0}~{\char5}------------{\char4}~~~{\char0}} \hbox{\amgr ~~~~~~~$\phantom {C_1(m)}$~~~~{\char5}------------------{\char4}} \hbox{\amgr ~~~~~~~$\phantom {C_1(m)}$~~~~~~~~~$l^*+2$~~~~}} \vskip 0.5 truecm \par\noindent \centerline {\smbfb Fig.5.3} \endinsert \par\noindent It can be easily shown that $H(C_1(m))$ is an increasing function of $m\in [l^*,m^*(l^*-1)-1]$, indeed $H(C_1(m+1))-H(C_1(m))= (h+\l)-2h\d>0$ by virtue of the hypothesis $\d<{h+\l\over 2h}$. Hence, $$H(C_{1}(m))\ge {\cal P}_1\;\; \forall m\in [l^*,m^*(l^*-1)-1] \;\; ;\Eq (5.4.1)$$ we observe that the equality is verified in $\equ (5.4.1)$ iff $m=l^*$, that is $C_1(m)\equiv {\cal P}_1$. \vskip 0.35 truecm \par\noindent {\it Case C2}. \par We consider the special saddles $C_{2,a}(l)$ and $C_{2,b}(l)$ in Fig.5.4. We remark that the configuration obtained from $C_{2,b}(l)$ by removing the protuberance is subcritical because $m^*(l-1)\ge m^*(l)+1$. \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,a}(l)}$~~~~{\char2}----------------{\char3}~~~~~~~~~~~~~~} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,a}(l)}$~~~~{\char0}~~~~{\char2}-{\char3}~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,a}(l)}$~~~~{\char0}~{\char2}--{\char4}~{\char5}-------{\char3}~{\char0}} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,a}(l)}$~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~{\char0}~~$m+3$} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,a}(l)}$~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,a}(l)}$~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~{\char0}~~~} \hbox{\amgr ~~~~~~~~~~~~$ {C_{2,a}(l)}$~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~${\widetilde L}\le l\le l^*{\char 45}1$} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,a}(l)}$~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~~~$m=m^*(l){\char 45}1$} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,a}(l)}$~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~{\char0}~~~~~~} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,a}(l)}$~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,a}(l)}$~~~~{\char0}~{\char5}------------{\char4}~{\char0}} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,a}(l)}$~~~~{\char5}----------------{\char4}} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,a}(l)}$~~~~~~~~~$l+2$~~~~~} \hbox{\amgr } \hbox{\amgr } \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,b}(l)}$~~~~{\char2}----------------{\char3}~~~~~~~~~~~~~~} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,b}(l)}$~~~~{\char0}~{\char2}----------{\char3}~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,b}(l)}$~~~~{\char0}~{\char0}~~~~~~~~~~{\char0}~~~{\char0}~~$m+2$} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,b}(l)}$~~~~{\char0}~{\char0}~~~~~~~~~~{\char0}~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,b}(l)}$~~~~{\char0}~{\char0}~~~~~~~~~~{\char0}~~~{\char0}~~~} \hbox{\amgr ~~~~~~~~~~~~$ {C_{2,b}(l)}$~~~~{\char0}~{\char0}~~~~~~~~~~{\char5}-{\char3}~{\char0}~~~~~~~~~~~~~~~${\widetilde L}\le l\le l^*{\char 45}1$} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,b}(l)}$~~~~{\char0}~{\char0}~~~~~~~~~~{\char2}-{\char4}~{\char0}~~~~~~~~~~~~~~~$m=m^*(l)$} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,b}(l)}$~~~~{\char0}~{\char0}~~~~~~~~~~{\char0}~~~{\char0}~~~~~~} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,b}(l)}$~~~~{\char0}~{\char0}~~~~~~~~~~{\char0}~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,b}(l)}$~~~~{\char0}~{\char5}----------{\char4}~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,b}(l)}$~~~~{\char5}----------------{\char4}} \hbox{\amgr ~~~~~~~~~~~~$\phantom {C_{2,b}(l)}$~~~~~~~~~$l+2$~~~~~}} \vskip 0.5 truecm \par\noindent \centerline {\smbfb Fig.5.4} \endinsert \par We have that $H(C_{2,a}(l))l$. \par We observe that $H(C_{2,a}(l))$ is a decreasing function of $l$: $$H(C_{2,a}(l+1))+1$, $m^*(l+1)-m^*(l)<-1$ and $m^*(l+1)-l^*+\d<0$ we obtain $H(C_{2,a}(l+1))-H(C_{2,a}(l))<(h+\l)-2h=\l-h<0$. This completes the proof of the inequality $\equ (5.5)$. \par Since $H(C_{2,a}(l))$ is a decreasing function, we have to compare the energy of the two configurations $C_{2,a}(l^*-1)$ and ${\cal P}_1$; by a direct calculation one obtains $H(C_{2,a}(l^*-1))>H({\cal P}_1)$. \vskip 0.35 truecm \par\noindent {\it Case B1}. \par We consider the special saddles ${\cal B}_{1,a}(M,{\hat M};{\hat L})$ and ${\cal B}_{1,b}({\hat M};L,{\hat L})$ in Fig.5.5 (here and in the following we use the notation introduced in Proposition 4.2 to label the internal and external sides; we use $L$, ${\hat L}$, $M$ and ${\hat M}$ to denote the dimensions of the birectangle obtained by removing the unit protuberance of the special saddle). \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~$\phantom {{\cal B}_{1,a}(M,{\hat M};{\hat L})}$~~~~{\char2}----------------{\char3}~~~~~~~~~{\char2}--------------{\char3}} \hbox{\amgr ~~~~~$\phantom {{\cal B}_{1,a}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~{\char0}~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~$\phantom {{\cal B}_{1,a}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~{\char2}------{\char3}~~~~~{\char0}~~~~~~~~~{\char0}~~{\char2}------{\char3}% ~~~~{\char0}} \hbox{\amgr ~~~~~$\phantom {{\cal B}_{1,a}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~{\char0}~~~~~~{\char0}~~~~~{\char0}~~~~~~~~~{\char0}~~{\char0}~~~~~~{\char0}% ~~~~{\char0}} \hbox{\amgr ~~~~~$\phantom {{\cal B}_{1,a}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~{\char0}~~~~~~{\char0}~~~~~{\char0}~~~~~~~~~{\char0}~~{\char0}~~~~~~{\char0}% ~~~~{\char0}} \hbox{\amgr ~~~~~$\phantom {{\cal B}_{1,a}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~{\char0}~~~~~~{\char5}-{\char3}~~~{\char0}~~~~~~~~~{\char0}~~{\char0}% ~~~~~~{\char0}~~~~{\char5}-{\char3}} \hbox{\amgr ~~~~~$\phantom {{\cal B}_{1,a}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~{\char0}~~~~~~{\char2}-{\char4}~~~{\char0}~~~~~~~~~{\char0}~~{\char0}% ~~~~~~{\char0}~~~~{\char2}-{\char4}} \hbox{\amgr ~~~~~$ {{\cal B}_{1,a}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~{\char0}~~~~~~{\char0}~~~~~{\char0}~~~~~~~~~{\char0}~~{\char0}~~~~~~{\char0}% ~~~~{\char0}~~~~${\cal B}_{1,b}({\hat M};L,{\hat L})$} \hbox{\amgr ~~~~~$\phantom {{\cal B}_{1,a}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~{\char0}~~~~~~{\char0}~~~~~{\char0}~~~~~~~~~{\char0}~~{\char0}~~~~~~{\char0}% ~~~~{\char0}} \hbox{\amgr ~~~~~$\phantom {{\cal B}_{1,a}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~{\char0}~~~~~~{\char0}~~~~~{\char0}~~~~~~~~~{\char0}~~{\char0}~~~~~~{\char0}% ~~~~{\char0}} \hbox{\amgr ~~~~~$\phantom {{\cal B}_{1,a}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~{\char5}------{\char4}~~~~~{\char0}~~~~~~~~~{\char0}~~{\char5}------{\char4}% ~~~~{\char0}} \hbox{\amgr ~~~~~$\phantom {{\cal B}_{1,a}(M,{\hat M};{\hat L})}$~~~~{\char5}----------------{\char4}~~~~~~~~~{\char5}--------------{\char4}} \hbox{\amgr ~~~~~~~~$\phantom {{\cal B}_{1,a}(M,{\hat M};{\hat L})}$~~~~~~~~~$M$~~~~~~~~~~~~~~~~~~~~~~~${\widetilde M}{\char 45}1$} \hbox{\amgr } \hbox{\amgr } \hbox{\amgr ~~~~~~~~$\phantom {{\cal B}_{1,a}(M,{\hat M};{\hat L})}$~~~~~~~~~$({\rm a})$~~~~~~~~~~~~~~~~~~~~~~~$({\rm b})$}} \vskip 0.5 truecm \par\noindent \centerline{ \vbox { \hsize=15truecm \baselineskip 0.35cm \noindent {\smbfb Fig.5.5}{\smb \quad (a) The internal horizontal dimension is $\scriptstyle {L^*-1}$ and the vertical one, $\scriptstyle {{\hat L}}$, is such that $\scriptstyle {{\hat L}\ge L^*}$. The external vertical dimension is $\scriptstyle {{\hat M}}$ and the horizontal one $\scriptstyle {M}$ is such that $\scriptstyle {{\widetilde M}\le Ml^*+2$ %%%%%% richiamo una formula della sezione 3 (see inequalities $\equ (3.14)$); %%%%%% fine richiamo \itemitem{$\bullet$} $R(L^*-1,L^*;l^*+2,l^*+2)\rightarrow R(L^*,L^*;l^*+2,l^*+2)$, $\D H_3<0$ because a whole internal slice of lenght $L^*$ has been attached to the internal (relatively) supercritical rectangle; \itemitem{$\bullet$} $R(L^*,L^*;l^*+2,l^*+2)\rightarrow R(l^*-1,l^*;l^*+2,l^*+2)$, $\D H_4<0$ because the internal rectangle is supercritical and $L^*H({\cal P}_1)$. This inequality and $\equ (5.7)$ lead us to the conclusion that $$H({\cal B}_{1,a}(M,{\hat M};{\hat L}))> {\cal P}_1\Eq (5.8)$$ for every possible choice of ${\hat M}$, $M$ and ${\hat L}$. \par In order to carachterize the special saddle ${\cal B}_{1,b}({\hat M};L,{\hat L})$, we have to distinguish two possible cases. \par\noindent Case (i) ${\hat L}+2\ge {\widetilde M}$: the birectangle $R(L,{\hat L};{\widetilde M}-1,{\hat M})$, obtained from ${\cal B}_{1,b}({\hat M};L,{\hat L})$ by removing the external unit protuberance, must be subcritical. Then, by virtue of Proposition 4.2, one can say that it must necessarily be $({\widetilde M}-1)-2\le l^*-1$, that is ${\widetilde M}\le l^*+2$. This is an absurd %%%%%% richiamo una formula della sezione 3 (see inequalities $\equ (3.14)$). %%%%%% fine richiamo Then we can conclude that it does not exist a special saddle ${\cal B}_{1,b}({\hat M};L,{\hat L})$ such that ${\hat L}+2\ge {\widetilde M}$. \par\noindent Case (ii) ${\hat L}+2< {\widetilde M}$: the internal rectangle $L\times {\hat L}$ must be contained in the rectangle $L\times (m^*(L)-1)$, otherwise the birectangle $R(L,{\hat L};{\widetilde M}-1,{\hat M})$ would be supercritical. Now we transform the special saddle ${\cal B}_{1,b}({\hat M};L,{\hat L})$ into $C_{2,a}(L+1)$ (notice that $L\ge L^*\Rightarrow L+1\ge {\widetilde L}$) and we show that the energy lowers. \itemitem{$\bullet$} ${\cal B}_{1,b}({\hat M};L,{\hat L})\rightarrow R(L,{\hat L};{\widetilde M}-1,{\hat M})$, $\D H_1=-[2J-(h-\l)]$; \itemitem{$\bullet$} $R(L,{\hat L};{\widetilde M}-1,{\hat M})\rightarrow R(L+1,{\hat L};{\widetilde M}-1,{\hat M})$, $\D H_2\le 0$ because ${\hat L}\ge L\ge L^*$; \itemitem{$\bullet$} $R(L+1,{\hat L};{\widetilde M}-1,{\hat M})\rightarrow R(L+1,m^*(L+1)-1;{\widetilde M}-1,{\hat M})$, $\D H_3\le 0$ because $L\ge L^*$ and ${\hat L}\le m^*(L+1)-1$; \itemitem{$\bullet$} $R(L+1,m^*(L+1)-1;{\widetilde M}-1,{\hat M})\rightarrow R(L+1,m^*(L+1)-1;L+2,m^*(L+1)+2)$, $\D H_4<0$ because the external rectangle is subcritical and $L+2<{\widetilde M}-1$; \itemitem{$\bullet$} $R(L+1,m^*(L+1)-1;L+2,m^*(L+1)+2)\rightarrow C_{2,a}(L+1)$, $\D H_5=2J-(h+\l)$. \par\noindent Hence, $$H({\cal B}_{1,b}({\hat M};L,{\hat L}))>H(C_{2,a}(L+1))>H({\cal P}_1) \Eq (5.9)$$ for every possible choice of the dimensions ${\hat M}$, $L$ and ${\hat L}$. \vskip 0.35 truecm \par\noindent {\it Case B2}. \par We consider the special saddle ${\cal B}_2({\hat M};L,{\hat L})$ in Fig.5.6. Two possible cases must be considered. \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_2({\hat M};L,{\hat L})}$~~~~{\char2}----------------{\char3}~~~~~~~~~~~~~~} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_2({\hat M};L,{\hat L})}$~~~~{\char0}~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_2({\hat M};L,{\hat L})}$~~~~{\char0}~{\char2}-------{\char3}~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_2({\hat M};L,{\hat L})}$~~~~{\char0}~{\char0}~~~~~~~{\char0}~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_2({\hat M};L,{\hat L})}$~~~~{\char0}~{\char0}~~~~~~~{\char0}~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_2({\hat M};L,{\hat L})}$~~~~{\char0}~{\char0}~~~~~~~{\char0}~~~~~~{\char5}-{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$ {{\cal B}_2({\hat M};L,{\hat L})}$~~~~{\char0}~{\char0}~~~~~~~{\char0}~~~~~~{\char2}-{\char4}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_2({\hat M};L,{\hat L})}$~~~~{\char0}~{\char0}~~~~~~~{\char0}~~~~~~{\char0}~~~~${\hat M}$~~~~~~~~~~~} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_2({\hat M};L,{\hat L})}$~~~~{\char0}~{\char0}~~~~~~~{\char0}~~~~~~{\char0}~~~~~~} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_2({\hat M};L,{\hat L})}$~~~~{\char0}~{\char0}~~~~~~~{\char0}~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_2({\hat M};L,{\hat L})}$~~~~{\char0}~{\char5}-------{\char4}~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_2({\hat M};L,{\hat L})}$~~~~{\char5}----------------{\char4}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_2({\hat M};L,{\hat L})}$~~~~~~~~~$M^*{\char 45}1$~~~~~}} \vskip 0.5 truecm \par\noindent \centerline{ \vbox { \hsize=15truecm \baselineskip 0.35cm \noindent {\smbfb Fig.5.6}{\smb \quad The internal dimensions $\scriptstyle {L}$ and $\scriptstyle {{\hat L}}$ are such that the birectangle obtained by removing the zero unit protuberance is subcritical. The external dimensions $\scriptstyle {M^*-1}$ and $\scriptstyle {M}$ are such that $\scriptstyle {{\hat M}\ge M^*}$.} } } \endinsert \par\noindent Case (i) $M^*>{\widetilde M}$: the internal rectangle is subcritical, hence by removing it we obtain a configuration at lower energy. Then by means of arguments similar to those used in the case of standard Ising model (see e.g. [NS1]), one can prove that $$H({\cal B}_2({\hat M};L,{\hat L}))\ge H({\cal P}_2)\;\; ,\Eq (5.6)$$ where the equality stands iff ${\cal B}_2({\hat M};L,{\hat L})\equiv {\cal P}_2$. \par\noindent Case (ii) $M^*={\widetilde M}$: see the discussion about the special saddle ${\cal B}_{1,b}({\hat M};L,{\hat L})$. \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{3}(M,{\hat M};{\hat L})}$~~~~{\char2}----------------{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{3}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{3}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~{\char2}------{\char3}~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{3}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~{\char0}~~~~~~{\char0}~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{3}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~{\char0}~~~~~~{\char0}~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{3}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~{\char0}~~~~~~{\char5}-{\char3}~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$ {{\cal B}_{3}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~{\char0}~~~~~~{\char2}-{\char4}~~~{\char0}~~~${\hat M}$} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{3}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~{\char0}~~~~~~{\char0}~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{3}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~{\char0}~~~~~~{\char0}~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{3}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~{\char0}~~~~~~{\char0}~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{3}(M,{\hat M};{\hat L})}$~~~~{\char0}~~~{\char5}------{\char4}~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{3}(M,{\hat M};{\hat L})}$~~~~{\char5}----------------{\char4}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{3}(M,{\hat M};{\hat L})}$~~~~~~~~~~$M$~~~~}} \vskip 0.5 truecm \par\noindent \centerline{ \vbox { \hsize=15truecm \baselineskip 0.35cm \noindent {\smbfb Fig.5.7}{\smb \quad The internal horizontal dimension is $\scriptstyle {l^*-1}$ and the vertical one $\scriptstyle {{\hat L}}$ is such that $\scriptstyle {l^*\le {\hat L}< {\widetilde M}-2}$ and $\scriptstyle {{\hat L}< m^*(l^*-1)}$. The external vertical dimension is $\scriptstyle {{\hat M}}$ and the horizontal one is $\scriptstyle {M<{\widetilde M}}$.} } } \endinsert \par \vskip 0.35 truecm \par\noindent {\it Case B3}. \par We consider, now, the special saddle ${\cal B}_{3}(M,{\hat M};{\hat L})$ in Fig.5.7. One can easily prove that $H({\cal B}_{3}(M,{\hat M};{\hat L}))\ge H(C_1({\hat L}))$ by virtue of the inequalities $MH({\cal P}_1)$ and $H({\cal B}_{4,b}(M,{\hat M};L))>H({\cal P}_1)$. \vskip 0.35 truecm \par\noindent {\it Case B5}. \par We consider the special saddle ${\cal B}_{5}({\hat M};L,{\hat L})$ in Fig.5.9. \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{5}({\hat M};L,{\hat L})}$~~~~{\char2}--------------{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{5}({\hat M};L,{\hat L})}$~~~~{\char0}~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{5}({\hat M};L,{\hat L})}$~~~~{\char0}~~{\char2}------{\char3}~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{5}({\hat M};L,{\hat L})}$~~~~{\char0}~~{\char0}~~~~~~{\char0}~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{5}({\hat M};L,{\hat L})}$~~~~{\char0}~~{\char0}~~~~~~{\char0}~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{5}({\hat M};L,{\hat L})}$~~~~{\char0}~~{\char0}~~~~~~{\char0}~~~~{\char5}-{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{5}({\hat M};L,{\hat L})}$~~~~{\char0}~~{\char0}~~~~~~{\char0}~~~~{\char2}-{\char4}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~$ {{\cal B}_{5}({\hat M};L,{\hat L})}$~~~~{\char0}~~{\char0}~~~~~~{\char0}~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{5}({\hat M};L,{\hat L})}$~~~~{\char0}~~{\char0}~~~~~~{\char0}~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{5}({\hat M};L,{\hat L})}$~~~~{\char0}~~{\char0}~~~~~~{\char0}~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{5}({\hat M};L,{\hat L})}$~~~~{\char0}~~{\char5}------{\char4}~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{5}({\hat M};L,{\hat L})}$~~~~{\char0}~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{5}({\hat M};L,{\hat L})}$~~~~{\char0}~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{5}({\hat M};L,{\hat L})}$~~~~{\char5}--------------{\char4}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~$\phantom {{\cal B}_{5}({\hat M};L,{\hat L})}$~~~~~~~~~$l^*+1$~~~~~}} \vskip 0.5 truecm \par\noindent \centerline{ \vbox { \hsize=15truecm \baselineskip 0.35cm \noindent {\smbfb Fig.5.9}{\smb \quad The internal horizontal dimension $\scriptstyle {L}$ is such that $\scriptstyle {L^*\le L\le l^*-1}$ and the vertical one $\scriptstyle {{\hat L}}$ is such that $\scriptstyle {{\widetilde M}-2\le {\hat L}< m^*(l^*-1)}$. The external vertical dimension is $\scriptstyle {{\hat M}}$. We remark that this special saddle does not exist, if we choose the parameters $\scriptstyle {h}$ and $\scriptstyle {\l}$ such that $\scriptstyle {m^*(l^*-1)\le {\widetilde M}-2}$.} } } \endinsert \par Now we transform the special saddle ${\cal B}_{5}({\hat M};L,{\hat L})$ into $C_{1}({\hat L})$ and we show that the energy lowers. \itemitem {$\bullet$} ${\cal B}_{5}({\hat M};L,{\hat L})\rightarrow R(L,{\hat L};l^*+2,{\hat M})$, $\D H_1=-(h-\l)({\hat M}-1)$; \itemitem {$\bullet$} $R(L,{\hat L};l^*+2,{\hat M})\rightarrow R(l^*-1,{\hat L};l^*+2,{\hat M})$, $\D H_2\le 0$ because $L\ge {\widetilde M}-2>L^*$ and $L\le l^*-1$; \itemitem {$\bullet$} $R(l^*-1,{\hat L};l^*+2,{\hat M})\rightarrow R(l^*-1,{\hat L};l^*+2,{\hat L}+2)$, $\D H_3\le 0$ since $l^*+2 H(C_{1}({\hat L}))\ge H({\cal P}_1)$. \vskip 0.35 truecm \par\noindent {\it Case B6}. \par We consider the special saddles ${\cal B}_{6,a}(M,{\hat M};L)$ and ${\cal B}_{6,b}(M,{\hat M};L)$ in Fig.5.10. \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~$\phantom {{\cal B}_{6,a}(M,{\hat M};L)}$~~~~{\char2}----------------{\char3}~~~~~~~~~{\char2}----------------{\char3}} \hbox{\amgr ~~$\phantom {{\cal B}_{6,a}(M,{\hat M};L)}$~~~~{\char0}~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~$\phantom {{\cal B}_{6,a}(M,{\hat M};L)}$~~~~{\char0}~~~~~{\char2}-{\char3}~~~~~~~~{\char0}~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~{\char0}% } \hbox{\amgr ~~$\phantom {{\cal B}_{6,a}(M,{\hat M};L)}$~~~~{\char0}~~~{\char2}-{\char4}~{\char5}--{\char3}~~~~~{\char0}~~~~~~~~~{\char0}~~{\char2}% ------{\char3}~~~~~~{\char0}} \hbox{\amgr ~~$\phantom {{\cal B}_{6,a}(M,{\hat M};L)}$~~~~{\char0}~~~{\char0}~~~~~~{\char0}~~~~~{\char0}~~~~~~~~~{\char0}~~{\char0}~~~~~~{\char0}% ~~~~~~{\char0}} \hbox{\amgr ~~$\phantom {{\cal B}_{6,a}(M,{\hat M};L)}$~~~~{\char0}~~~{\char0}~~~~~~{\char0}~~~~~{\char0}~~~~~~~~~{\char0}~~{\char0}~~~~~~{\char0}% ~~~~~~{\char0}} \hbox{\amgr ~~$\phantom {{\cal B}_{6,a}(M,{\hat M};L)}$~~~~{\char0}~~~{\char0}~~~~~~{\char0}~~~~~{\char0}~~~~~~~~~{\char0}~~{\char0}~~~~~~{\char0}~~~~~~{\char5}-{\char3}} \hbox{\amgr ~~$ {{\cal B}_{6,a}(M,{\hat M};L)}$~~~~{\char0}~~~{\char0}~~~~~~{\char0}~~~~~{\char0}~~~~~~~~~{\char0}~~{\char0}~~~~~~{\char0}~~~~~~{\char2}-{\char4}~~~~${\cal B}_{6,b}(M,{\hat M};L)$} \hbox{\amgr ~~$\phantom {{\cal B}_{6,a}(M,{\hat M};L)}$~~~~{\char0}~~~{\char0}~~~~~~{\char0}~~~~~{\char0}~~~~~~~~~{\char0}~~{\char0}~~~~~~{\char0}% ~~~~~~{\char0}} \hbox{\amgr ~~$\phantom {{\cal B}_{6,a}(M,{\hat M};L)}$~~~~{\char0}~~~{\char0}~~~~~~{\char0}~~~~~{\char0}~~~~~~~~~{\char0}~~{\char0}~~~~~~{\char0}% ~~~~~~{\char0}} \hbox{\amgr ~~$\phantom {{\cal B}_{6,a}(M,{\hat M};L)}$~~~~{\char0}~~~{\char0}~~~~~~{\char0}~~~~~{\char0}~~~~~~~~~{\char0}~~{\char0}~~~~~~{\char0}% ~~~~~~{\char0}} \hbox{\amgr ~~$\phantom {{\cal B}_{6,a}(M,{\hat M};L)}$~~~~{\char0}~~~{\char5}------{\char4}~~~~~{\char0}~~~~~~~~~{\char0}~~{\char5}------{\char4}% ~~~~~~{\char0}} \hbox{\amgr ~~$\phantom {{\cal B}_{6,a}(M,{\hat M};L)}$~~~~{\char0}~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~$\phantom {{\cal B}_{6,a}(M,{\hat M};L)}$~~~~{\char0}~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~$\phantom {{\cal B}_{6,a}(M,{\hat M};L)}$~~~~{\char5}----------------{\char4}~~~~~~~~~{\char5}----------------{\char4}} \hbox{\amgr ~~$\phantom {{\cal B}_{6,a}(M,{\hat M};L)}$~~~~~~~~~~~$M$~~~~~~~~~~~~~~~~~~~~~~~~$M$}} \vskip 0.5 truecm \par\noindent \centerline{ \vbox { \hsize=15truecm \baselineskip 0.35cm \noindent {\smbfb Fig.5.10}{\smb \quad (a) The internal horizontal dimension $\scriptstyle {L}$ is such that $\scriptstyle {L\ge L^*}$. The vertical one is $\scriptstyle {{\hat L}=m^*(M-2)-1}$ and it is such that $\scriptstyle {{\hat L}+3\ge {\widetilde M}}$. The external vertical dimension is $\scriptstyle {{\hat M}}$ and the horizontal one $\scriptstyle {M}$ is such that $\scriptstyle {{\widetilde L}\le M-2\le l^*-1}$. (b) The internal horizontal dimension $\scriptstyle {L}$ is such that $\scriptstyle {L\ge L^*}$. The vertical one is $\scriptstyle {{\hat L}=m^*(M-1)}$ and it is such that $\scriptstyle {{\hat L}+2\ge {\widetilde M}}$. The external vertical dimension is $\scriptstyle {{\hat M}}$ and the external horizontal dimension $\scriptstyle {M}$ is such that $\scriptstyle {{\widetilde L}\le M-1\le l^*-1}$. We remark that for certain choices of the parameters $\scriptstyle {h}$ and $\scriptstyle {\l}$ the configurations in (a) and (b) cannot be considered.} } } \endinsert \par Now, we transform the special saddle ${\cal B}_{6,a}(M,{\hat M};L)$ into $C_{2,a}(M-2)$ and show that the energy lowers. \itemitem{$\bullet$} ${\cal B}_{6,a}(M,{\hat M};L)\rightarrow {\cal B}_{6,a}(M,{\hat L}+3;L)$, $\D H_1\le 0$ since $M\le l^*+1L^*$. \par\noindent Hence, we conclude that $H({\cal B}_{6,a}(M,{\hat M};L))\ge H(C_{2,a}(M-2))>H({\cal P}_1)$. \par The special saddle ${\cal B}_{6,b}(M,{\hat M};L)$ can be transformed into the configuration $C_{2,b}(M-1)$ lowering the energy. \itemitem{$\bullet$} ${\cal B}_{6,b}(M,{\hat M};L)\rightarrow R(L,{\hat L};M+1,{\hat M})$, $\D H_1=-(h-\l)({\hat M}-1)$; \itemitem{$\bullet$} $R(L,{\hat L};M+1,{\hat M})\rightarrow R(L,{\hat L};M+1,{\hat L}+2)$, $\D H_2\le 0$ since $M+1\le l^*+1L^*$; \itemitem{$\bullet$} $R(M-2,{\hat L};M+1,{\hat L}+2)\rightarrow C_{2,b}(M-1)$, $\D H_4=2J-(h+\l)$. \par\noindent It is easily seen that $\D H_1+\D H_4<0$, hence $H({\cal B}_{6,a}(M,{\hat M};L))>H(C_{2,b}(M-1))>H({\cal P}_1)$. \par This completes the proof of Lemma 5.1 in the case $\d<{h+\l\over 2h}$. We suppose, now, $\d>{h+\l\over 2h}$ and observe that in this case $$m^*(l^*-1)=l^*\;\; ,\Eq (5.12)$$ as it follows from equation $\equ (5.4)$. In the sequel we will analyze all the cases that have to be discussed with arguments different from those used before. \vskip 0.35 truecm \par\noindent {\it Case C1}. \par The special saddle $C_1(m)$ with $l^*\le m\le m^*(l^*-1)-1$ cannot be considered, since $m^*(l^*-1)-1=l^*-1$ (see $\equ (5.12)$). \vskip 0.35 truecm \par\noindent {\it Case C2}. \par We proved above that $C_{2,a}(l^*-1)$ is the special saddle with lowest energy among $C_{2,a}(L)$ and $C_{2,b}(L)$. This result is not dependent on the value of the real number $\d$. Hence, one can say $H(C_{2,b}(l))>H(C_{2,a}(l))\ge H(C_{2,a}(l^*-1)) =H({\cal P}_1)$ (we remark that in the case $\d>{h+\l\over 2h}$ the special saddle $C_{2,a}(l^*-1)$ and the global saddle ${\cal P}_1$ coincide). \vskip 0.35 truecm \par\noindent {\it Case B1}. \par In order to prove that $H({\cal B}_{1,a}(M,{\hat M};{\hat L}))>H({\cal P}_1)$ we have to consider two different cases. \par\noindent Case (i) ${\hat L}\ge l^*-1$: we transform the special saddle ${\cal B}_{1,a}(M,{\hat M};{\hat L})$ into ${\cal P}_1$ and we prove that the energy lowers. \itemitem{$\bullet$} ${\cal B}_{1,a}(M,{\hat M};{\hat L})\rightarrow R(L^*-1,{\hat L};M,{\hat M})$, $\D H_1=-[2J-(h+\l)]$; \itemitem{$\bullet$} $R(L^*-1,{\hat L};M,{\hat M})\rightarrow R(L^*-1,l^*-1;M,{\hat M})$, $\D H_2\le 0$ since $L^*-1l^*+2$; \itemitem{$\bullet$} $R(l^*-1,l^*-1;l^*+2,l^*+1)\rightarrow {\cal P}_1$, $\D H_5=2J-(h+\l)$. \par\noindent We conclude that $H({\cal B}_{1,a}(M,{\hat M};{\hat L}))>H({\cal P}_1)$ since $\sum_{1=1}^5 \D H_i<0$. \par\noindent Case (ii) $L^*\le {\hat L}L^*$. Now we transform ${\cal B}_{1,a}(M,{\hat M};{\hat L})$ into ${\cal P}_1$. \itemitem{$\bullet$} ${\cal B}_{1,a}(M,{\hat M};{\hat L})\rightarrow R(L^*-1,{\hat L};M,{\hat M})$, $\D H_1=-[2J-(h+\l)]$; \itemitem{$\bullet$} $R(L^*-1,{\hat L};M,{\hat M})\rightarrow R(l^*-1,{\hat L};M,{\hat M})$, $\D H_2< 0$ since $L^*-1l^*+2$; \itemitem{$\bullet$} $R(l^*-1,l^*-1;l^*+2,l^*+1)\rightarrow {\cal P}_1$, $\D H_5=2J-(h+\l)$. \par\noindent Also in this case we conclude that $H({\cal B}_{1,a}(M,{\hat M};{\hat L}))>H({\cal P}_1)$. \par Finally, with arguments similar to those used in the case $\d<{h+\l\over 2h}$ one can show that $H({\cal B}_{1,b}({\hat M};L,{\hat L}))>H(C_{2,a}(L))>H({\cal P}_1)$. \vskip 0.35 truecm \par\noindent {\it Case B3}. \par This case cannot be considered because the inequalities $l^*\le {\hat L}\t_{\piuuno})=1\;\; .\Eq (6.1')$$ \par The crucial property of ${\cal G}$ will be that the minimum of the energy in its boundary $\partial {\cal G}$ will be given by ${\cal P}_1$ or ${\cal P}_2$. \par We will see that this implies that for every configuration $\sigma$ with sufficiently low energy ($H(\sigma)<\min\{ H({\cal P}_1), H({\cal P}_2) \}$) $\equ (6.1)$ is verified. \par Let us now give an example of a configuration belonging to ${\cal G}$ which is potentially supercritical in the sense that $\equ (6.1)$ fails. \par Consider an acceptable configuration $\eta$ which is different from $\menouno$ only in a square $\L_0:=\L_{L_0}$ ($\eta (x)=-1\; \forall x\in \L\setminus\L_0$); the even integer $L_0$ will be chosen later on. \par Consider the four sublattices of spacing $2$ into which ${\bf Z}^{2}$ is partitioned and write $${\bf Z}^{2}\equiv {\bf Z}^{2}_1= {\bf Z}^{2}_{2,a}\cup {\bf Z}^{2}_{2,b}\cup {\bf Z}^{2}_{2,c}\cup {\bf Z}^{2}_{2,d}$$ where the subscript $1$ or $2$ denotes the spacing; $a,\; b,\; c,\; d$ label the four sublattices of spacing $2$. \par Suppose that ${\bf Z}^{2}_{2,b}$ and ${\bf Z}^{2}_{2,d}$ belong to the same sublattice of spacing $\sqrt 2$ and set $$\eta (x)=\left\{ \eqalign{ +1& \;\;\; \forall x\in {\bf Z}^{2}_{2,a} \cap \L_0\cr -1& \;\;\; \forall x\in {\bf Z}^{2}_{2,b} \cap \L_0\cr -1& \;\;\; \forall x\in {\bf Z}^{2}_{2,d} \cap \L_0\cr 0& \;\;\; \forall x\in {\bf Z}^{2}_{2,c} \cap \L_0\cr}\right. \;\; ;$$ (see Fig.6.1). \par Starting from our definition of ${\cal G}$ we have first to transform the $-1$ with a plus spin among its nearest neighbours into $0$. In this way we get the configuration $\eta_1$ depicted in the left hand side of Fig.6.1. Eventually, we get a configuration $\hat\eta$ with a unique plurirectangle with external edges with length $L_0+1$ and many non--interacting unit squares of pluses in its interior. \par On the other hand, starting from $\eta$ we can change the $-1$ in $\L_0$ with two plus spins among their nearest neighbours into $+1$ by decreasing the energy; we obtain the configuration $\eta_2$ depicted in the right hand side of Fig.6.1. Subsequently, still decreasing the energy the configuration $\eta_2$ can be transformed into the configuration $\eta^*=C(L_0-1,L_0-1)$. \par Now, if $L_0$ is chosen such that $l^*+1\le L_0~~~+~+~+~+~+~+~+~{\char45}} \hbox{\amgr ~~~~~~~~~~~~~0~0~0~0~0~0~0~0~~~~~~~~{\char45}~0~{\char45}~0~{\char45}~0~{\char45}~0~~~~~~~~+~0~+~0~+~0~+~0} \hbox{\amgr ~~~~~~~~~~~~~+~0~+~0~+~0~+~0~~~~~~~~+~{\char45}~+~{\char45}~+~{\char45}~+~{\char45}~~~~~~~~+~+~+~+~+~+~+~{\char45}} \hbox{\amgr ~~~~~~~~~~~~~0~0~0~0~0~0~0~0~~~~~~~~{\char45}~0~{\char45}~0~{\char45}~0~{\char45}~0~~~~~~~~{\char45}~0~{\char45}~0~{\char45}~0~{\char45}~0}} \vskip 0.5 truecm \par\noindent \centerline {\smbfb Fig.6.1} \endinsert \par To construct ${\cal G}$, first of all we define a map ${\cal F}:\s\rightarrow {\hat \s}={\cal F}\s$ with $\s$ an acceptable configuration and ${\hat \s}$ a local minimum of the energy, such that the two following properties are satisfied $$\eqalign{ H({\hat \s})&\le H(\s)\cr \s & \prec {\hat \s}\cr}\;\; ;\Eq (6.2)$$ that is the local minimum ${\hat \s}$ is bigger than $\s$ and at a lower energy level. Then we define the set ${\cal G}$ as the set of configurations $\s$ such that ${\hat \s}$ is subcritical, that is $P_{\hat \s} (\t_{\menouno}<\t_{\piuuno})\rightarrow 1$ as $\b\rightarrow\infty$. \par Now we define the map ${\cal F}:\s\rightarrow {\hat \s}$; the definition is given in the following five steps. Let $\s$ be an acceptable configuration: \par $(i)$ starting from $\s$ we construct the configuration $\s_1$ by turning into zero all the minus spins of $\s$ which have at least one plus spin among their nearest neighbour sites. We remark that $H(\s_1)\le H(\s)$ (see Fig.3.1) and $\s\prec\s_1$. \par $(ii)$ Let us denote by $c^-_1$ the minus spins cluster in the configuration $\s_1$ which is winding around the torus and by $c^-_i$ all the other minus spins clusters in $\s_1$. In $\s_1$ there is no direct interface $+-$, then we can conclude that every $c^-_i$ cluster is inside a zero spins cluster (see Fig.6.2). Now we consider the configuration $\s_2$ obtained from $\s_1$ by turning into zero all the minus spins in all the clusters $c^-_i$. The result $\s_1\prec\s_2$ is obvious. We have, also, that $H(\s_2)\le H(\s_1)$; indeed in every cluster $c^-_i$ there is at least one minus spin with two zero spins among its nearest neighbours; this spin can be transformed into zero lowering the energy. We can repeat this argument until all the spins of the starting cluster $c^-_i$ have been transformed into zero. \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char2}-----{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~0~~{\char5}-------{\char3}~~~~~{\char2}----{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~{\char2}-------{\char4}~~~~~~~~~~~~~{\char0}~~~{\char2}-{\char4}~~~~{\char5}-{\char3}% } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~0~~~{\char0}~~~{\char0}~~~0~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~0~~~{\char2}----{\char3}~~~~~~~~{\char0}~{\char2}-{\char4}~~{\char2}-{\char3}% ~~~{\char5}---{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~{\char2}-----{\char4}~~~~{\char2}--{\char4}~~~~{\char5}---{\char3}~~~~{\char5}-{\char4}~~~~% {\char0}~{\char5}-{\char3}~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~{\char0}~~~~{\char45}~~~~~~{\char0}~~~~~~~~~{\char2}-{\char4}~~~{\char0}~~% 0~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~{\char5}--{\char3}~~~0~~~{\char5}--{\char3}~~~~{\char45}~~~{\char0}~~~0~~~{\char2}-{\char4}~% ~{\char45}~~{\char5}-{\char3}~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~{\char0}~~~~~~~~{\char0}~~~~~{\char2}-{\char4}~~~~~~~~~{\char0}~~~{\char5}% -{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char5}----{\char3}~~0~~{\char5}--{\char3}~~~{\char2}-{\char4}~~~~~{\char0}~~~{\char45}~% ~~~{\char45}~~{\char0}~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~{\char5}---{\char4}~~~~~{\char2}-{\char4}~~~~~~~~~~~{\char0}~~~{\char2}% -{\char4}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char5}---{\char3}~~~~~~~~~~~~~~{\char0}~~~{\char45}~~~{\char2}-----{\char4}~~~{\char0}% } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~0~~~~~~{\char5}-------{\char4}~~~~~~0~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char5}---------{\char3}~~~~~~~~~~~~~~~{\char2}------{\char4}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~0~~~~{\char2}-----{\char4}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char5}---------{\char4}}} \vskip 0.5 truecm \par\noindent \centerline {\smbfb Fig.6.2} \endinsert \par $(iii)$ In $\s_2$ there is no direct interface $+-$, then we observe that every cluster of plus spins is inside a cluster of zero spins; it can happen that in some of the plus spins clusters there are one or more clusters of zero spins (see Fig.6.3). We construct the configuration $\s_3$ by removing all these clusters of zero spins. With arguments similar to those used in step $(ii)$ one can prove that $H(\s_3)\le H(\s_2)$ and $\s_2\prec\s_3$. \par $(iv)$ The configuration $\s_3$ is made of a minus spins cluster which is winding around the torus, the zero spins clusters denoted by $c^{0}_{i}\;\forall i\in\{1,2,...,k^{0}\}$ and the clusters with plus spins $c^{+}_{i,j}\;\forall i\in\{1,2,...,k^{0}\}$ and $\forall j\in\{1,2,...,k^{+}_{i}\}$. The clusters $c^{+}_{i,j}\;\forall j\in\{1,2,...,k^{+}_{i}\}$ are all inside the cluster $c^{0}_{i}$. We consider, now, the rectangular envelopes $R^{0}_{i}=R(c^{0}_{i})\;\forall i\in\{1,2,...,k^{0}\}$ and the configuration $\s_4$ obtained by filling all these rectangles with zero spins; in this step the plus spins are not changed. It is immediate that $H(\s_4)\le H(\s_3)$ and $\s_3\prec\s_4$. \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char2}-----{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~0~~{\char5}-------{\char3}~~~~~{\char2}----{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~{\char2}-------{\char4}~~~~~~~{\char2}--{\char3}~~{\char0}~~~{\char2}-{\char4}~~~~{\char5}% -{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~{\char2}-----{\char4}~~{\char0}~~{\char0}~~~{\char0}~~~0~~~~{\char0}% } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~0~~~{\char2}-{\char4}~~~+~~~~{\char0}~~{\char0}~{\char2}-{\char4}~~{\char2}% -{\char3}~~~{\char5}---{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~{\char2}-----{\char4}~~~~{\char2}--{\char4}~~~~{\char2}---{\char3}~{\char0}~~{\char5}-{\char4}% ~~~~{\char0}~{\char5}-{\char3}~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~{\char0}+~~~~{\char2}-{\char4}~~~{\char0}~{\char0}~~~~~~~{\char2}-{\char4}% ~~~{\char0}~~0~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~{\char5}--{\char3}~~~0~~~{\char5}--{\char3}~~{\char0}~~~~~{\char0}~{\char5}-------{\char4}~% ~+~~{\char5}-{\char3}~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~{\char0}~~{\char0}~~0~~{\char5}-----{\char3}~~~~~{\char2}--{\char3}~~{\char0}% ~~~{\char5}-{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~{\char5}----{\char3}~~0~~{\char0}~~{\char5}-----------{\char4}~~{\char2}--{\char4}~0{\char0}% ~~{\char0}~~0~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~{\char5}------{\char3}~{\char2}---{\char3}~~~~{\char5}-----{\char4}~~{\char0}% ~~~{\char2}-{\char4}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char5}---{\char3}~~~~~~~~{\char5}-{\char4}~~~{\char0}~~+~~~~{\char2}-----{\char4}% ~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~0~~~~~~{\char5}-------{\char4}~~~~~~0~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char5}---------{\char3}~~~~~~~~~~~~~~~{\char2}------{\char4}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~0~~~~{\char2}-----{\char4}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char5}---------{\char4}}} \vskip 0.5 truecm \par\noindent \centerline {\smbfb Fig.6.3} \endinsert \par $(v)$ Apart from the plus spins cluster, the configuration $\s_4$ is made of zero rectangular clusters placed in the ``sea" of minus spins. We obtain the configuration $\s_5$ by means of the {\it chain construction} used in [KO1], applied to the rectangular clusters $R^{0}_{i}\;\forall i\in\{1,2,...,k^{0}\}$. \par Let us briefly describe this construction. Given a set of rectangles $R^0_1,\dots ,R^0_l$ we partition it into maximal connected components ${\cal C}^{(1)}_j$ with $j=1,\dots ,k^{(1)}$ called {\it chains of first generation} $$(R^0_1\dots R^0_l)=({\cal C}^{(1)}_1\dots {\cal C}^{(1)}_{k^{(1)}})\;\; .$$ The notion of connection is given by pairwise interaction: a set $R^0_1,\dots ,R^0_m$ of rectangles is connected if it cannot be divided into two non--interacting parts. \par Now consider the $k^{(1)}$ rectangles $R({\cal C}^{(1)}_j)$ obtained as rectangular envelope of the union of the rectangles belonging to ${\cal C}^{(1)}_j$. Partition this set of rectangles into maximal connected component: in this way we construct the chains of second generation ${\cal C}^{(2)}_1,\dots ,{\cal C}^{(2)}_{k^{(2)}}$. We continue in this way up to a finite maximal order $n$ such that the chains of the $n$--th generation are non--interacting rectangles (see [KO1] for more details). \par We call $\sigma_5$ this configuration containing these non--interacting rectangular clusters ${\bar R}^{0}_{i}\;\forall i\in\{1,2,...,k^{0,f}\}$ of zero spins placed in the minus spins ``sea". With usual arguments one can prove that $H(\sigma_{5})\leq H(\sigma_{4})$ and $\s_4\prec\s_5$. \par $(vi)$ By repeating the operations described in points $(iv)$ and $(v)$ for the plus spins clusters lying in every rectangle ${\bar R}^{0}_{i}\;\forall i\in\{1,2,...,k^{0,f}\}$, we obtain the final configuration ${\hat \s}$. This configuration is made of the external rectangular zero spins clusters ${\bar R}^{0}_{i}\;\forall i\in\{1,2,...,k^{0,f}\}$ and the internal non-interacting plus spins clusters ${\bar R}^{+}_{i,j}\;\forall i\in\{1,2,...,k^{0,f}\}$ and $\forall j\in\{1,2,...,k^{+,f}_i\}$. As usual one can prove that $H({\hat \s})\leq H(\sigma_{5})$ and $\s_5\prec {\hat \s}$. \par The definition of the map ${\cal F}$ is now complete, we observe that ${\hat \s}$ is a local minimum and that the properties $\equ (6.2)$ are satisfied. Finally we remark that the map ${\cal F}$ is {\it monotone} in the sense that $$\sigma\prec\eta\;\Rightarrow\; {\hat\sigma}\prec{\hat\eta}\;\; , \Eq (6.3)$$ for every couple of acceptable configurations $\s$ and $\eta$. \par Now we state the following %%%%%%%%%%%%%%%%%Proposition 6.1 \vskip 0.35 truecm \noindent {\bf Proposition 6.1.}\par\noindent $$U({\cal G})\subset {\hat {\cal P}}\;\; . \Eq (6.4)$$ Namely the set of minima of the energy in the boundary of ${\cal G}$ is contained in ${\hat {\cal P}}$. %%%%%%%%%%%%%%%%%%%End of Proposition 6.1 \par\noindent {\it Proof.} \vskip 0.5 truecm \par In order to prove Proposition 6.1 we consider a configuration $\eta\in\partial {\cal G}$ and we show that there exists a special saddle $\tilde\eta$ such that $H(\eta)\ge H(\tilde\eta)$. Then Proposition 6.1 will follow from Lemma 5.1. \par Let us consider $\eta\in {\cal G}$; there exists a configuration $\sigma =\eta ^{x,b}$ with $x\in \L$ and $b\not= \eta (x)$ such that $\sigma\in {\cal G}$. By virtue of the monotonicity of the map ${\cal F}$ (see equation $\equ (6.3)$) and of the fact that ${\hat \sigma}$ is a subcritical local minimum, it follows that $b < \eta (x)$; hence we also have that $b\not= +1$. \par We denote by $R^0_i({\hat\s})\;\forall i\in\{ 1,...,k^0({\hat\s})\}$ and by $R^+_{i,j}({\hat\s})\;\forall j\in\{ 1,...k^+_i({\hat\s})\}$ and $\forall i\in\{ 1,...,k^0({\hat\s})\}$ the rectangles respectively of zeros and pluses which appear in the configuration ${\hat\s}$; we remark that all the rectangles $R^+_{i,j}({\hat\s})\;\forall j\in\{ 1,...,k^+_i({\hat\s})\}$ are inside the zero rectangle $R^0_i({\hat\s})$. In the following, by abuse of notation, we will also denote by $R^0_i({\hat\s})$ what we will call {\it structure} $R^0_i({\hat\s})$, namely the complex given by the ``external" rectangle togheter with all its ``internal" rectangles of pluses (what before we called plurirectangle is indeed a configuration containing a unique structure). \vskip 0.5 truecm \par\noindent {\it Case 1}: $b=-1$ and $\eta (x)=0$. \par >From the definition of the map ${\cal F}$ easily follows that necessarily $x$ lies outside the rectangles $R^0_i({\hat\s})$. \par Given the configuration ${\hat\eta}$ we denote by $R^0_i({\hat\eta})\;\forall i\in\{ 1,...,k^0({\hat\eta})\}$ and by $R^+_{i,j}({\hat\eta})\;\forall j\in\{ 1,...,k^+_i({\hat\eta})\}$ and $\forall i\in\{ 1,...,k^0({\hat\eta})\}$ the rectangles respectively of zeros and pluses which appear in it. We denote by ${\bar R}^0({\hat\eta})$ the supercritical structure among the $R^0_i({\hat\eta})$ and by ${\bar R}^0_1,...,{\bar R}^0_s$ the rectangles of zeros such that $\forall i\in\{ 1,...,s\}\;{\bar R}^0_i$ appears in ${\hat\sigma}$ and $\forall i\in\{ 1,...,s\}\;{\bar R}^0_i$ is ``inside" the rectangle ${\bar R}^0({\hat\eta})$. \par We consider, now, the configuration $\eta_1$ defined as follows: $\eta_1(x)=0$, all the other spins are minus except for the zeros and the pluses of the structures ${\bar R}^0_i\;\forall i\in\{ 1,...,s\}$. It can be easily proved that $H(\eta)\ge H(\eta_1)$. We distinguish the two cases $1.1$ and $1.2$. \vskip 0.3 truecm \par\noindent {\it Case 1.1}: all the rectangles of pluses which appear in $\eta_1$ are subcritical. \par We consider the configuration $\eta_{1.1}$ obtained from $\eta_1$ by changing into zeros all the plus spins. We remark that $H(\eta_1)\ge H(\eta_{1.1})$ because $\eta_{1.1}$ has been constructed by removing subcritical rectangles of pluses. \par With an Ising--like argument (see e.g. [KO1]) one can prove that $H(\eta_{1.1})\ge H({\cal P}_2)$. Hence in the case $1.1$ we have found a special saddle with energy lower than the starting configuration $\eta$. \vskip 0.3 truecm \par\noindent {\it Case 1.2}: in $\eta_1$ there exists at least one supercritical rectangle of pluses. \par We consider the configuration $\eta_{1.2}$ obtained from $\eta_1$ by removing in every structure ${\bar R}^0_i\;\forall i\in\{ 1,...,s\}$ all the subcritical rectangles of pluses and by filling with pluses the rectangular envelope of the union of the supercritical rectangles of pluses. We remark that every structure ${\bar R}^0_i\;\forall i\in\{ 1,...,s\}$ in $\eta_{1.2}$ is either ``empty" (with no rectangle of pluses inside) or it has just a rectangle of pluses inside and this rectangle is supercritical. \par We denote by $Q$ the unit square centered at the site $x\in\L$; we distinguish the two following cases: \vskip 0.3 truecm \par\noindent {\it Case 1.2.1}: one of the structures ${\bar R}^0_i\;\forall i\in\{ 1,...,s\}$ of $\eta_{1.2}$ (we denote it by ${\bar R}^0_{1.2.1}$) interacts with $Q$ and the structure obtained by filling with zeroes the rectangular envelope of ${\bar R}^0_{1.2.1}\cup Q$ is supercritical. \par Let us denote by $\eta_{1.2.1}$ the configuration obtained by removing in $\eta_1$ all the structures ${\bar R}^0_i\;\forall i\in\{ 1,...,s\}$ except for ${\bar R}^0_{1.2.1}$. If $Q$ is adjacent to ${\bar R}^0_{1.2.1}$ then $\eta_{1.2.1}$ is a special saddle. Otherwise $Q$ is at distance one from one of the sides of the rectangle ${\bar R}^0_{1.2.1}$ or $Q$ and ${\bar R}^0_{1.2.1}$ touch in a corner; in this case it can be esily found a special saddle with energy lower than $H(\eta_{1.2})$. \par Hence in the case $1.2.1$ a special saddle with energy lower than the starting configuration $\eta$ has been found. \vskip 0.3 truecm \par\noindent {\it Case 1.2.2}: the condition $1.2.1$ is not fulfilled. \par By an argument similar to the one used in [KO1] (see pages 1136--1137 therein) we can find two structures ${\tilde R}_1$ and ${\tilde R}_2$ such that: they are both subcritical, their external rectangles are interacting, the structure obtained by filling with zeroes their rectangular envelope is supercritical and $H(\eta_{1.2})\ge H({\tilde R}_1)+H({\tilde R}_2)$ (when we say $H({\tilde R}_i)\;$ with $i\in\{ 1,2\}$ we are referring to the energy of the configuration obtained by plunging the structure ${\tilde R}_i$ in the ``sea" of minuses). We still have to distinguish between two possible cases. \vskip 0.3 truecm \par\noindent {\it Case 1.2.2.1}: both structures $H({\tilde R}_i)\;$ with $i\in\{ 1,2\}$ have a supercritical rectangle of pluses inside. \par Now we consider a just--supercritical frame whose external rectangle is contained in the rectangular envelope of the union of the two external rectangles of ${\tilde R}_1$ and of ${\tilde R}_2$. Such a frame surely exists and we denote it by ${\tilde C}$. \par Starting from ${\tilde R}_1$ and ${\tilde R}_2$ and recalling that these structures are subcritical, one can construct two other structures, ${\tilde \Re}_1$ and ${\tilde \Re}_2$ (birectangles or frames), such that the three following conditions are satisfied: $i)$ $H({\tilde R}_1)\ge H({\tilde \Re}_1)$ and $H({\tilde R}_2)\ge H({\tilde \Re}_2)$; $ii)$ if the two external rectangles of the two structures ${\tilde \Re}_1$ and ${\tilde \Re}_2$ touch by a corner then the rectangular envelope of the union of the external rectangles of ${\tilde \Re}_1$ and of ${\tilde \Re}_2$ coincides exactly with the external rectangle of the frame ${\tilde C}$; $iii)$ at least one of the two internal rectangles of pluses (the one in ${\tilde \Re}_1$ or the one in ${\tilde \Re}_2$) is supercritical. \par If one considers the configuration $\eta_{1.2.2.1}$ obtained by plunging the structures in the ``sea" of minus spins such that the external rectangles of ${\tilde \Re}_1$ and of ${\tilde \Re}_2$ touch by a corner, one can easily convince himself that $H(\eta_{1.2})\ge H(\eta_{1.2.2.1})$. \par Finally, starting from $\eta_{1.2.2.1}$ we construct the special saddle ${\tilde\eta}$ by performing the following steps: $i)$ we fill of zeroes the rectangular envelope of the union of the two external rectangles of zeroes in $\eta_{1.2.2.1}$; $ii)$ we let grow the internal supercritcal rectangle of pluses until the frame ${\tilde C}$ is reached; $iii)$ we transform into zeroes all the pluses, except for one, of one of the four sides of the internal rectangle, such that a special saddle is obtained. It can be easily proved that $H(\eta_{1.2.2.1}) > H({\tilde \eta})$ by comparing the energy differences involved in the three steps described above. We remark that the energy increase of the third step is largely compensated by the energy decrease involved in the second step. \vskip 0.3 truecm \par\noindent {\it Case 1.2.2.2}: one of the structures $H({\tilde R}_i)$ is ``empty", in the sense that it has no rectangles of pluses inside. \par This case can be discussed with arguments similar to those used in the Case 1.2.2.2. \vskip 0.5 truecm \par\noindent {\it Case 2}: $b=-1$ and $\eta (x)=+1$. \par Starting from ${\hat\s}$ one can always construct a configuration $\eta_2$ such that: $i)$ $\eta_2\in\partial {\cal G}$; $ii)$ $\exists y\in\L$ such that $\eta_2 (y)=0$ and $\eta_2^{y,-1}\in {\cal G}$. In this way the proof has been reduced to the Case 1. \vskip 0.5 truecm \par\noindent {\it Case 3}: $b=0$. \par The site $x$ is inside one of the rectangles of zeroes $R^0_i({\hat\s})\;\forall i\in\{ 1,...,k^0({\hat\s})\}$; we denote it by ${\bar R}^0$. There are two possible cases that must be considered. \vskip 0.3 truecm \par\noindent {\it Case 3.1}: $x$ is not on one of the boundary slices of ${\bar R}^0$ (the tipical situation is depicted in Fig.6.4). \par In this case the rectangles of zeroes in ${\hat\eta}$ coincide with those in ${\hat\sigma}$, but the structure ${\bar R}^0({\hat\eta})$ is supercritical (${\bar R}^0({\hat\eta})$ is the structure of ${\hat\eta}$ such that its external rectangle of zeroes coincides with ${\bar R}^0$). \par We consider, now, the configuration $\eta_{3.1}$ defined as follows: $i)$ $\eta_{3.1}$ is obtained starting from ${\hat\sigma}$, by removing all the structures $R^0_i({\hat\s})\;\forall i\in\{ 1,...,k^0({\hat\s})\}$ except for the one whose external rectangle coincides with the external rectangle of the structure ${\bar R}^0$ (we denote this structure by ${\bar R}^0({\hat\sigma})$); $ii)$ $\eta_{3.1}(x)=+1$. It can be easily proved that $H(\eta)\ge H(\eta_{3.1})$. \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char2}---------------------------------------{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~{\char2}--------{\char3}~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char2}------------{\char3}~~~~~~~~{\char0}~~~~~~~~{\char0}~~~~~~{\char0}% } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~~~~~~~~{\char5}--------{\char4}~~~~~~{\char0}% } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}x~~~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char5}------------{\char4}~~~{\char2}--------{\char3}~~~~~~~~~~~{\char0}% } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~{\char0}~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~{\char0}~~~~~{\char2}--{\char3}~~{\char0}% } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~{\char5}--------{\char4}~~~~~{\char0}~~{\char0}~~{\char0}% } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~{\char0}~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char5}--{\char4}~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char5}---------------------------------------{\char4}}} \vskip 0.5 truecm \par\noindent \centerline {\smbfb Fig.6.4} \endinsert \par We denote by $R_1$ the rectangular envelope of the union of the supercritical rectangles of pluses inside ${\bar R}^0({\hat\sigma})$ and by $R_2$ the rectangular envelope of the union of the supercritical rectangles of pluses inside ${\bar R}^0({\hat\eta})$. We remark that the two sctructures ${\bar R}^0({\hat\sigma})$ and ${\bar R}^0({\hat\eta})$ have different internal rectangles of pluses, even though their external rectangles of zeroes coincide. \par Now we observe that there exists a rectangle $R_3$ contained in $R_2$ and containing $R_1$ such that the configuration with all the spins minus except for the zeroes in the rectangle ${\bar R}^0$ and the pluses in $R_3$ is an almost--supercritical configuration. We consider the special saddle ${\tilde\eta}$ obtained by properly putting a unit plus protuberance to one of the four sides of the internal rectangle of pluses of the almost--supercritical configuration found before. It can be easily shown that $H(\eta_{3.1})\ge H({\tilde\eta})$. Hence, even in this case, we have found a special saddle with energy lower than the energy of the starting configuration $\eta\in\partial {\cal G}$. \vskip 0.3 truecm \par\noindent {\it Case 3.2}: $x$ is on one of the boundary slices of ${\bar R}^0$ (see, for example, Fig.6.5). \par We construct the configuration $\eta_{3.2}$ starting from ${\hat\sigma}$ and by turning into zero only the spin minus at a site nearest neighbour to $x$. One can easily convince himself that $H({\hat\eta})\ge H(\eta_{3.2})$. If $\eta_{3.2}\in\partial {\cal G}$ then the proof is reduced to Case 1; if $\eta_{3.2}\in {\cal G}$ the proof is reduced to Case 3.1. \par The proof of Proposition 6.1 is now complete. $\square$ \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char2}---------------------------------------{\char3}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~{\char2}--------{\char3}~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char2}------------{\char3}~~~~~~~~{\char0}~~~~~~~~{\char0}~~~~~~{\char0}% } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~~~~~~~~{\char5}--------{\char4}~~~~~~{\char0}% } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~{\char5}------------{\char4}~~~{\char2}--------{\char3}~~~~~~~~~~~{\char0}% } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~{\char0}~~~~~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~{\char0}~~~~~{\char2}--{\char3}~~{\char0}% } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~{\char5}--------{\char4}~~~~~{\char0}~~{\char0}~x{\char0}% } \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~{\char0}~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char0}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\char5}--{\char4}~~{\char0}} \hbox{\amgr ~~~~~~~~~~~~~~~~~~~~~~~~{\char5}---------------------------------------{\char4}}} \vskip 0.5 truecm \par\noindent \centerline {\smbfb Fig.6.5} \endinsert \par \vfill\eject \numsec=7\numfor=1 {\bf Section 7. Proof of the theorems.} \par Let us first give some definitions extending the ones given in Section 4.\par We recall that by $ C ( l_1, l_2)$ we denote the set of configurations containing only a frame with internal sides $ l_1, l_2$. We recall the notation $ l := \min \{l_1,\; l_2\} , \; m := \{l_1,\; l_2\}$. \par We denote by $ S(l_1, l_2)$ the set of configurations obtained from $ C (l_1, l_2)$ by substituting one of the smaller internal sides with a unit square protuberance namely by substituting all but one plus spins adjacent from the interior to one of the internal sides of length $l$ with zeroes (see Fig.7.1). \par We denote by $R(l_1, l_2)$ the set of configurations containing a unique birectangle obtained by erasing the internal unit square protuberance from $ S(l_1, l_2)$ (see Fig.7.1). We denote by $ G(l_1, l_2)$ the set of configurations obtained from the frame $C (l_1, l_2)$ by adding a unit square spin $0$ protuberance to one of the longer external sides in $C (l_1, l_2)$. A particularly relevant case will be the one $ | l_1-l_2 | \leq 1$ where either $ m = l + 1 $ or $ m = l $. We remark that $ G(l - 1, l )$ is obtained from the birectangle $ R(l, l)$ by substituting one ``free" external row (or column) of zeroes of length $l+2$ with a unit square protuberance (see Fig.7.1); similarly $ G(l - 1, l-1 )$ is obtained from $ R(l - 1, l )$ by substituting one free external row or column of spin 0 of length $l+1$ with a unit square protuberance. \par Finally let us denote by $\bar R(l_1,l_2) := R(0,0, l_1,l_2)\cup R(0,0, l_2,l_1)$ the set of configurations without plus spins where the zero spins are precisely the ones contained inside a rectangle with sides equal, respectively, to $l_1,l_2$. \midinsert \vskip 0.5 truecm \vbox{\font\amgr=AMGR at 10truept\baselineskip0.1466667truein\lineskiplimit-\maxdimen \catcode`\-=\active\catcode`\~=\active\def~{{\char32}}\def-{{\char1}}% \hbox{\amgr ~~~~~~~~~~~~~~~$C(l,l)$~~~~~~~~~~~~~~~~~~$S(l,l)$~~~~~~~~~~~~~~~~~$R(l,l)% $} \hbox{\amgr ~~~~~~~~{\char2}-----------------{\char3}~~~~{\char2}-----------------{\char3}~~~~{\char2}-----------% ------{\char3}} \hbox{\amgr ~~~~~~~~{\char0}~{\char2}-------------{\char3}~{\char0}~~~~{\char0}~{\char2}-----------{\char3}~~~{\char0}% ~~~~{\char0}~{\char2}-----------{\char3}~~~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~{\char0}~{\char0}~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~~~{\char0}% ~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~~~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~{\char0}~{\char0}~~~~{\char0}~{\char0}~~~~~~~~~~~{\char5}-{\char3}% ~{\char0}~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~~~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~{\char0}~{\char0}~~~~{\char0}~{\char0}~~~~~~~~~~~{\char2}-{\char4}% ~{\char0}~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~~~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~{\char0}~{\char0}~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~~~{\char0}% ~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~~~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~{\char0}~{\char0}~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~~~{\char0}% ~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~~~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~~~{\char0}~{\char0}~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~~~{\char0}% ~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~~~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char5}-------------{\char4}~{\char0}~~~~{\char0}~{\char5}-----------{\char4}~~~{\char0}% ~~~~{\char0}~{\char5}-----------{\char4}~~~{\char0}} \hbox{\amgr ~~~~~~~~{\char5}-----------------{\char4}~~~~{\char5}-----------------{\char4}~~~~{\char5}-----------% ------{\char4}} \hbox{\amgr ~~~~~~~~~~~~~~~~$l+2$~~~~~~~~~~~~~~~~~~~$l+2$~~~~~~~~~~~~~~~~~~~$l+2$% } \hbox{\amgr } \hbox{\amgr } \hbox{\amgr } \hbox{\amgr ~~~~~~~~~~~~~$G(l{\char45}1,l)$~~~~~~~~~~~~~$C(l{\char45}1,l)$~~~~~~~~~~~~~~$% S(l{\char45}1,l)$} \hbox{\amgr ~~~~~~~~{\char2}---------------{\char3}~~~~~~{\char2}---------------{\char3}~~~~~~{\char2}-----------% ----{\char3}} \hbox{\amgr ~~~~~~~~{\char0}~{\char2}-----------{\char3}~{\char0}~~~~~~{\char0}~{\char2}-----------{\char3}~{\char0}% ~~~~~~{\char0}~{\char2}-----------{\char3}~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}% ~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}% ~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char5}-{\char3}~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}% ~{\char0}~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char2}-{\char4}~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}% ~{\char0}~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}% ~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}% ~~~~~~{\char0}~{\char5}---{\char3}~{\char2}-----{\char4}~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char5}-----------{\char4}~{\char0}~~~~~~{\char0}~{\char5}-----------{\char4}~{\char0}% ~~~~~~{\char0}~~~~~{\char5}-{\char4}~~~~~~~{\char0}} \hbox{\amgr ~~~~~~~~{\char5}---------------{\char4}~~~~~~{\char5}---------------{\char4}~~~~~~{\char5}-----------% ----{\char4}} \hbox{\amgr ~~~~~~~~~~~~~~$l+1$~~~~~~~~~~~~~~~~~~~$l+1$~~~~~~~~~~~~~~~~~~~$l+1$% } \hbox{\amgr } \hbox{\amgr } \hbox{\amgr } \hbox{\amgr ~~~~~~~~~~~~~$R(l{\char45}1,l)$~~~~~~~~~~~~$G(l{\char45}1,l{\char45}1)$~~~~~~~~~~% ~$C(l{\char45}1,l{\char45}1)$} \hbox{\amgr ~~~~~~~~{\char2}---------------{\char3}~~~~~~{\char2}---------------{\char3}~~~~~~{\char2}-----------% ----{\char3}~~~~} \hbox{\amgr ~~~~~~~~{\char0}~{\char2}-----------{\char3}~{\char0}~~~~~~{\char0}~{\char2}-----------{\char3}~{\char0}% ~~~~~~{\char0}~{\char2}-----------{\char3}~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}% ~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}% ~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}% ~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}% ~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}% ~~~~~~{\char0}~{\char0}~~~~~~~~~~~{\char0}~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~{\char5}-----------{\char4}~{\char0}~~~~~~{\char0}~{\char5}-----------{\char4}~{\char0}% ~~~~~~{\char0}~{\char5}-----------{\char4}~{\char0}} \hbox{\amgr ~~~~~~~~{\char0}~~~~~~~~~~~~~~~{\char0}~~~~~~{\char5}------{\char3}~{\char2}------{\char4}~~~~~~{\char5}% ---------------{\char4}} \hbox{\amgr ~~~~~~~~{\char5}---------------{\char4}~~~~~~~~~~~~~{\char5}-{\char4}} \hbox{\amgr ~~~~~~~~~~~~~~$l+1$~~~~~~~~~~~~~~~~~~~$l+1$~~~~~~~~~~~~~~~~~~~$l+1$}} \vskip 0.8 truecm \par\noindent \centerline{ \vbox { \hsize=13truecm \baselineskip 0.35cm \noindent {\bf Fig.7.2}{\rm \quad Contraction of a squared frame. The energy differences involved in each single step of the contraction are: $\scriptstyle {(h+\lambda)(l-1)}$, $\scriptstyle {-[2J-(h+\lambda)]}$, $\scriptstyle {(h-\lambda)(l+1)}$, $\scriptstyle {-[2J-(h-\lambda)]}$, $\scriptstyle {(h+\lambda)(l-2)}$, $\scriptstyle {-[2J-(h+\lambda)]}$, $\scriptstyle {(h-\lambda)l}$, $\scriptstyle {-[2J-(h-\lambda)]}$.} } } \endinsert \par We want to prove now Theorem 1. \par Let ${\cal P}$ be the set of protocritical saddles or special minimal saddles. \par If $ 0 < 2 \l < h $: ${\cal P} = {\cal P}_2$ in Fig.5.1; namely ${\cal P}$ is the set of configurations with no pluses and a unique cluster of zeroes given by a rectangle with sides $M^*, M^*-1$ with a unit square protuberance attached to one of its longer sides.\par If $0 < \l < h <2\l$ and $\d < { h+\l \over 2h}$ then ${\cal P}= {\cal P}_{1,a} := S(l^*,l^*)$. If $ 0 < \l { h+\l \over 2h}$ then ${\cal P}= {\cal P}_{1,b} := S(l^*-1,l^*)$ (see Fig.5.1). \par Now we notice that the set ${\cal G} \subset\O _{\L}$, defined in Section 6 satisfies the following properties:\par\noindent 1. ${\cal G}$ connected; $\menouno \; \in \; {\cal G},\;\;\; \piuuno \;\not \in \; {\cal G}$. \par\noindent 2. There exists a path $\o : \menouno \to {\cal P} $, contained in ${\cal G}$, with $$ H(\s) < H({\cal P}) \;\;\; \forall \s\in \o , \;\;\s \not= {\cal P} \Eq (7.0a) $$ and there exists a path $\o' :{\cal P} \to \piuuno $, contained in ${\cal G}^c$, with $$ H(\s) < H({\cal P}) \;\;\; \forall \s\in \o' ,\;\; \s \not= {\cal P}\;\; . \Eq (7.0b) $$ In the case ${\cal P}={\cal P}_1$ $\equ (7.0b)$ easily follows from the arguments of proof of Proposition 4.1: $\o$ is constructed following a sequence of shrinking subcritical droplets whereas $\o'$ follows a sequence of growing supercritical droplets. In the case ${\cal P}={\cal P}_2$ $\equ (7.0b)$ follows from the arguments of proof of Proposition 4.2. \par\noindent 3. The minimal energy in $\partial {\cal G}$ is attained only for ``protocritical'' (global saddle) configurations $\sigma \in {\cal P}$; namely, $$ \min_{\sigma\in \partial {\cal G}}(H(\sigma)-H(-\underline 1))= H({\cal P})-H(-\underline 1)=: \;\Gamma\;\; , \Eq (7.0) $$ $$ \min_{\sigma\in \partial {\cal G}\setminus {\cal P}}(H(\sigma)-H({\cal P})) \;>\; 0\;\; . \Eq (7.0') $$ \bigskip We notice that, starting from any $\sigma\in{\cal P}$, we can change a spin adjacent to the unit square protuberance always present in ${\cal P}$ (from $-1$ to $0$ in ${\cal P}_2$ if $h>2\l$ and from $0$ to $ +1$ in ${\cal P}_{1,a}$ or ${\cal P}_{1,b}$ if $h<2\l$) in order to get a ``stable protuberance of length 2". This protuberance is called stable since its growth takes place decreasing the energy while its shrinking requires an increase of energy. The probability of the above described single spin change is not smaller than $ { 1\over |\L|}$ (see, for instance, [NS1] for more details on this point). \par In other words, with probability separated from zero, uniformly in $\b$, starting from ${\cal P}$, we reach the strict basin of attraction of a supercritical minimum. Then, for any $\varepsilon>0$, it follows from Proposition 4.1 that the probability to reach $+\underline 1$ before reaching $-\underline 1$, can be bounded from below as: $$ P_{{\cal P}}(\tau_{+\underline 1}<\tau_{-\underline 1})\geq \exp (-\e \b)\;\; . \Eq (7.0'') $$ We get from Proposition 4.1 that, for $\b$ sufficiently large, the typical time, starting from ${\cal P}_1$ to reach $\piuuno$ is much shorter than the typical time to get to ${\cal P}$ starting from $\menouno$ $$ \lim_{\beta\to\infty}P_{{\cal P}_1}(\tau_{+\underline 1}<\exp(\G_1) \mid \tau_{+\underline 1}<\tau_{-\underline 1}) =1\;\; . \Eq (7.0''') $$ for a suitable $\G_1 \; < \; \G$.\par Moreover by an analysis totally analogous to the one needed for the Ising model (see for instance [NS1]) one can get the same results starting from ${\cal P}_2$; namely $$ \lim_{\beta\to\infty}P_{{\cal P}_2}(\tau_{+\underline 1}<\exp(\G_2) \mid \tau_{+\underline 1}<\tau_{-\underline 1}) =1\;\; . \Eq (7.0'''') $$ for a suitable $\G_2 \; < \; \G$.\par In Appendix A we state and prove a result concerning the sequence of passages through ${\cal P}$ and the typical time to see an ``efficient" passage through ${\cal P}$ namely one followed by a descent to $\piuuno$. \par >From Propositions 3.4, 3.7 in [OS1], Proposition A.1 of Appendix A, \equ (7.0''), \equ (7.0''') we easily get Theorem 1. $\square$ \par\bigskip We want now to give the definition of the tube ${\cal T}$ of trajectories appearing in the statement of Theorem 2 below. It represents the typical mechanism of escape from metastability in the sense that, with probability tending to 1 as $\b$ tends to infinity, during its first excursion from $\menouno$ to $\piuuno$, our process will follow a path in ${\cal T}$.\par ${\cal T}$ will be optimal in the sense that it cannot be really reduced without loosing in probability.\par ${\cal T}$ involves a sequence of ``droplets" with suitable geometric shapes and suitable ``resistance times" in some ``permanence sets" of configurations related to these droplets. The precise statement about the typical paths during the first excursion between $\menouno$ and $\piuuno$ will involve a certain randomness of these resistance times inside the different permanence sets appearing in ${\cal T}$.\par In ${\cal T}$ we will distinguish two parts. The ``up" part ${\cal T}_u$ namely the ascent to ${\cal P}$ and the ``down" part ${\cal T}_d$ from ${\cal P}$ to $\piuuno$. This second part ${\cal T}_d$ is {\it almost downhill} in the sense that all the paths $\o = \s_0,\s_1, \dots ,\s_i , \dots \in \; {\cal T}_d$ will be such that: $$ \s_0 ={\cal P}, \; \exists \;\;\bar T \; : \; \s _{\bar T} \; = \; \piuuno ,\;\;\;\;\; \max _{ \s \in \o \setminus {\cal P} } H(\s)\; < \; H({\cal P}), \;\;\; \min_{\s \in \o} H(\s) = H(\piuuno )\;\; . $$ Whereas ${\cal T}_u$ is {\it almost uphill} in the sense that all the paths $\o = \s_0,\s_1, \dots ,\s_i , \dots \in \; {\cal T}_u$ will be such that: $$ \s_0 =\menouno, \; \exists \;\;\bar T \; : \; \s _{\bar T} \; = \; {\cal P} ,\;\;\;\;\; \max _{ \s \in \o \setminus {\cal P} }H(\s) \; < \; H({\cal P}), \;\;\; \min_{\s \in \o} H(\s) = H(\menouno )\;\; . $$ In the following we give the definition of the time reversed tube $\bar {\cal T} $ of ${\cal T}_u$.\par $\bar {\cal T}$ will also be almost downhill; it will describe the typical first ``descent" from the protocritical saddle to $\menouno$. By general arguments based on reversibility (see ref. [S1]), we will deduce the desired results on the first excursion from $\menouno$ to ${\cal P}$ saying that with probability tending to one as $\b$ tends to $\infty$ it takes place in the tube ${\cal T}_u$. Then to conclude our construction of ${\cal T}$ we will only have to determine ${\cal T}_d$. \par Let us now recall some basic definitions of [OS1] concerning the first descent from any configuration $\h_0$ contained in a given cycle $A$ to the bottom $F(A)$ valid not only for our Blume-Capel Metropolis dynamics but also for a general ``low temperature" Markov chain satisfying Hypothesis M in Appendix A. We refer to Appendix A where this more general set-up is introduced. \par We will first define in general the set of ``standard cascades" emerging from a configuration $\h _0$; our intention is to apply a (simplified version of a) result of [OS1] telling that with high probability when $\b \to \infty$ the first descent from $\h _0$ to $F(A)$ follows, in a well specified way, a standard cascade. Thus the main model-dependent work will be to determine, in our specific case, the set of standard cascades. In particular we will reduce the problem of the determination of the tube of typical trajectories followed by our process during its first descent to $\menouno$ starting from a configuration $\s_0$ in ${\cal G}$ immediately reached starting from the global saddle ${\cal P}$, along a downhill path entering ${\cal G}$, to find the set (denoted by $\bar {\cal T})$ of all the standard cascades (in a suitable cycle) emerging from $\s_0$. \par \bigskip A {\it standard cascade} emerging from a state $\h_0$ is a sequence: $$ {\cal T} ( \h_0, \o_1, \h_1,\o_2, \dots , \h_{M-1}, \o_M)\;=\; \o_1\cup Q_1\cup \o_2, \dots , Q_{M-1}\cup \o_M \cup Q_M, \Eq (7.1) $$ where for $i=1,\dots, M$: $\o_i$ is a downhill path emerging from $\h_{i-1}$ and ending inside the ``permanence set" $Q_i$. Each path $\o_i$ can be downhill continued up to a stable equilibrium point $\x_i \in Q_i$. The $Q_i$'s are special sets being a sort of generalized cycles containing also the minimal saddles between $\x_i$ and $F(A)$; for $i=1,\dots, M-1$ $\h_{i} \in Q_i $ are minimal saddles between $\x_i$ and $F(A)$; finally $ \x_M \subseteq Q_M \subseteq F(A)$ (see [OS1], Section 4 for more details). \par Notice that $\o_i$ can just reduce to one downhill step from $\h _{i-1}$ to $Q_i$; in this case we use the convention: $\o_i = \h _{i-1} $. \par We do not give here the precise definition of the $Q_i$'s since it happens that we do not really need it. In our particular case of Metropolis dynamics for the Blume-Capel model with particular initial conditions (of interest for our applications) as we will check we have some semplifications w.r.t the general case.\par The most important is that the $Q_i$'s, for $i=1,\dots, M-1$ are replaced by genuine cycles $A_i$; $\h_i$, not contained in $A_i$, is an element of ${\cal S} (A_i)$ and $\o_i$ ends in the interior of $A_i$.\par We will apply the general theory developed in [OS1] to two cases. In the first one, when analyzing $\bar {\cal T}$ the cycle $A$ will be the maximal connected set $\bar A$ in $\O _{\L}$ containing $\menouno$ with energy less than $H({\cal P})$. It follows from Proposition 3.4 in [OS1] that $\bar A$ is contained in the set ${\cal G}$ introduced in Section 6 and that ${\cal S}(\bar A) \equiv {\cal P}$. Always in this case we have: $F(\bar A)\equiv Q_M \equiv \x_M \equiv \menouno$.\par In the second case, in the study of ${\cal T}_d$ the cycle $A$ will be the maximal connected component $\tilde A$ in $\O _{\L}$ containing $\piuuno$ with energy less than $H({\cal P})$. It is immediate to see that $\tilde A \subset {\cal G}^c$. In this case we have: $F(\tilde A)\equiv Q_M \equiv \x_M \equiv \piuuno$.\par In both cases, as we said before, for suitable initial conditions we will verify that the $Q_i$'s for $i=1,\dots, M-1$ are replaced by genuine cycles $A_i$; $M$ will depend on the initial configuration $\h_0$ as well as on the particular choice of the parameters $J,h,\l$. $\o_i$ ends in the interior of $A_i$; $\h_i \in {\cal S}(\x_i, F(A_{i+1}))$, not contained in $A_i$ as we said before are minimal saddles in the boundary $\partial A_i$. The cycles $A_i$ are precisely the maximal connected components containing $\x_i$ with energy less than $H(\h_i)$ ($\x_i\in A_i$ are the minima towards which $\o_i$ can be downhill continued). \par We consider an initial configuration $\h_0 $ corresponding to one of the following five cases:\par 1. $A = \bar A$: $\h_0 \in \bar A\cap [R(l,l)\cup R(l,l+1)]$ for some $l\ge \widetilde L$. \par 2. $A = \bar A$: $\h_0 \in \bar R(M^*,M^*-1)$.\par 3. $A = \tilde A$, $0 < \l < h <2\l$ and $\d > { h+\l \over 2h}$: $\h_0 \in \bar B (C(l^*-1,l^*))$.\par 4. $A = \tilde A$, $0 < \l < h <2\l$ and $\d < { h+\l \over 2h}$: $\h_0 \in \bar B (C(l^*,l^*) )$.\par 5. $A = \tilde A$: $\h_0 \in \bar B (\bar R(M^*,M^*))$. \par \vskip.5cm {\bf Remark.}\par We could even consider much more general initial configurations $\h _0$. It is not true (see the definition of the set $\bar D$ in Section 4 that for {\it any } $\h _0$ the simplified version (involving genuine cycles $A_i$ in place of the sets $Q_i$) of the general [OS1] results holds true. In fact with the very particular choice $\h_0 \in \bar A\cap [R(l,l)\cup R(l,l+1)]$ as we will see an even simpler statement holds: the $\o_i$ will be almost all coinciding with $\h _i$ (in the above specified sense).\par \vskip.5cm {\bf Warning.}\par We want to warn the reader of the use that we are going to make, in the construction of the tube ${\cal T}$, of the equivalence class of configurations as it has been specified in the remark in the proof of Proposition 4.1. In fact, strictly speaking what we will construct and call standard cascades are {\it sets} of standard cascades obtained from equivalence classes of configurations modulo rotations, translations, inversion and ``displacement of protuberances". \par \bigskip Let us now start with the definition of the set $\bar {\cal T}$ of the standard cascades emerging from a configuration $\s_0$ in ${\cal G}$ immediately reached starting from the global saddle ${\cal P}$, along a downhill path entering into ${\cal G}$. \par We consider first the case $a = h/\l <2$ . The other case of $a > 2$ is almost identical to the corresponding one for the Ising model and will be treated later on.\par We have to distinguish two cases: $\d \; = l^* - { 2J -(h-\l)\over h} <\; { h +\l \over 2h}$, when the global saddle ${\cal P}$ has the form ${\cal P}_{1,a} = S(l^*, l^*)$ given in Fig.5.1; or $\d \; >\; { h +\l \over 2h}$ when the global saddle has the form ${\cal P}_{1,b } = S(l^* - 1, l^*)$ also given in Fig.5.1. \par Let us first consider the case $\d \; < \; { h +\l \over 2h}$ (like in Fig.7.2 for $l=l^*$). Let $\bar {\cal P}_1 = R(l^*, l^*) $ be the configuration obtained from ${\cal P}_1$ by erasing the unit square protuberance. $\bar {\cal P}_1$ is a subcritical birectangle; it belongs to the set ${\cal G}$ and satisfies condition 1 above.\par To construct the tube $\bar {\cal T}$ we have basically to solve the above described sequence of minimax problems. To simplify the exposition we divide the tube $\bar {\cal T}$ into four segments corresponding to four different mechanisms of contraction; we write: $$ \bar {\cal T} = \bar {\cal T}_1 \cup \bar {\cal T}_2 \cup\bar {\cal T}_3 \cup \bar {\cal T}_4\;\; . \Eq (7.2') $$ The most relevant ones are the first and the second part. As we will see the third part for $h < 2\l$ reduces just to a simple downhill path. \par We start from the determination of the minimal saddle $\h_1 := {\cal S}(\bar {\cal P}_1,\menouno)$ between $\bar {\cal P}_1$ and $\menouno$.\par >From the results of Section 4 we know that ${\cal S}(\bar {\cal P}_1,\menouno)$ is not trivial in the sense that it differs from $\bar {\cal P}_1$ and $$ \h_1 := \; {\cal S}(\bar {\cal P}_1,\menouno) \; = S(l^*- 1, l^*) \;\; .\Eq (7.3)$$ Thus the first ``permanence set" $Q_1$ of our standard cascade is the cycle $A_{l^*-1, l^*}$ defined as the maximal connected set of configurations containing $R(l^*, l^*)$ with energy less than $H( S(l^*-1,l^*) )$. We recall that the basic inequality to be checked in order to get \equ (7.3) is $$H( S(l,l) ) - H( S(l-1,l) )> 0$$ which is verified for $L^*\leq l\leq l^*-1$.\par For any $l$: $L^* \leq l \leq l^* -1 $ we define the cycle $A_{l,l}\; (A_{l,l+1})$ as the maximal connected set of configurations containing $R(l,l) \; ( R(l,l+1) )$ with energy less than $H( S(l-1,l) )$ $(H( S(l,l)))$ (see Fig.7.2). By extending the previous argument we get that the first part of our standard cascade is given by: $$ \bar {\cal T}_1 = A_{l^*-1, l^*}, S(l^*-1,l^*),A_{l^*-1,l^*-1}, S(l^*-1,l^*-1),A_{l^*-2,l^*-1}, $$ $$\dots , S(\widetilde L,\widetilde L+1), A_{\widetilde L,\widetilde L}, S(\widetilde L,\widetilde L) \Eq (7.4)$$ Then we observe that for $l\leq \widetilde L -1$, we have $$\eqalign{ H( S(l,l) ) &< H( G(l,l) )\cr H( S(l,l+1) ) &< H( G(l,l+1) )\cr}\;\; ;\Eq (7.5) $$ \equ (7.5) are the basic inequalities to get that, for $l_0 \le l < \widetilde L$: $$\eqalign{ {\cal S}(R(l,l+1), \menouno) &= G(l,l)\cr {\cal S}(R(l,l), \menouno) &= G(l-1,l)\cr}\;\; .\Eq (7.6) $$ It is clear from the results of Section 4 that the subsequent permanence sets are the cycles: $$ A^1_{\widetilde L,\widetilde L -1},A^2_{\widetilde L,\widetilde L -1}, A^1_{\widetilde L-1,\widetilde L -1},A^2_{\widetilde L-1,\widetilde L -1}, \dots, A^1_{l,l},A^2_{l,l}, A^1_{l,l-1},A^2_{l,l-1},\dots , A^2_{l_0,l_0 } \Eq (7.7) $$ where $ l_0 =[{h \over \l} +1], \;\; l_0 \leq l$; we notice that for our present choice of the parameters: $\l < h < 2 \l$, we have $ l_0 =2$ but we could consider a general situation $ l_0 > 2$ as well when analizing the contraction of a subcritical frame in the region $h > 2\l$ (case 1 above). Moreover, for $l_0 \leq l\leq \widetilde L $: \par $A^1_{l,l-1}\; = $ maximal connected component of the set of configurations containing $R(l,l)$ with energy less than $H( G(l-1,l))$; namely $A^1_{l,l-1} $ is the strict basin of attraction of $R(l,l)$: $$ A^1_{l,l-1}\; = \bar B ( R(l,l))\;\; , $$ with bottom $$ F(A^1_{l,l-1}) \; = R(l,l) $$ and minimal saddle $$ {\cal S} (A^1_{l,l-1})\; = G (l-1,l)\;\; ; $$ $A^2_{l,l-1}\; = $ maximal connected component containing $C(l-1,l)$ with energy less than $H(S(l-1,l))$. We have: $$ A^2_{l,l-1}\; = \bar B ( C(l-1,l)) $$ $$ F(A^2_{l,l-1}) \; = C(l-1,l) $$ and $$ {\cal S} (A^2_{l,l-1})\; = S(l-1,l)\;\; . $$ For $l_0 + 1 < l \leq \widetilde L $ we define $A^1_{l-1,l-1} \; = $ maximal connected component containing $R(l-1,l)$ with energy less than $H( G(l-1,l-1))$. We have $$ A^1_{l-1,l-1}\; = \bar B ( R(l-1,l)) $$ $$ F(A^1_{l-1,l-1}) \; = R(l-1,l) $$ and $$ {\cal S} (A^1_{l-1,l-1})\; = G (l-1,l-1)\;\; . $$ $A^2_{l-1,l-1}\; = $ maximal connected component containing $C(l-1,l-1)$ with energy less than $H(S(l-1,l-1))$. We have: $$ A^2_{l-1,l-1}\; = \bar B ( C(l-1,l-1)) $$ $$ F(A^2_{l-1,l-1}) \; = C(l-1,l-1) $$ and $$ {\cal S} (A^2_{l-1,l-1})\; = S(l-1,l-1)\;\; . $$ Then the second segment of the standard cascade is: $$ A^1_{\widetilde L,\widetilde L -1}, G (\widetilde L -1,\widetilde L), A^2_{\widetilde L,\widetilde L -1}, S (\widetilde L -1,\widetilde L), A^1_{\widetilde L-1,\widetilde L -1},G (\widetilde L-1,\widetilde L -1), A^2_{\widetilde L-1,\widetilde L -1}, $$ $$ S (\widetilde L-1,\widetilde L -1),\dots, S(l_0,l_0 ) ,A^1_{l_0,l_0-1}\;\; . \Eq (7.8) $$ We notice that both the first and the second part $ \bar{\cal T} _1$, $ \bar{\cal T} _2$ of the tube $ \bar{\cal T}$ describe a contraction following squared or almost squared frames; but whereas in the first part the permanence sets are cycles with many minima in their interior, in the second part they are ``one well" in the sense that they coincide with the strict basin of attraction of their bottoms. The typical times spent inside these cycles and the typical states visited before leaving them are different in the two cases of $ \bar{\cal T} _1$ and $ \bar{\cal T} _2$.\par The third part $ \bar{\cal T} _3$, that we are going to define, corresponds to the shrinking of the interior rectangle of the frame $C(l_0,l_0)$. Indeed it follows from Section 4 (see \equ (4.14) therein) that for $ l < l_0$ the lowest minimal saddles in the boundary of the basin of attraction $A^1_{l_0,l_0-1 }$ of the birectangle $ R(l_0,l_0) \equiv R( l_0,l_0-1, l_0+2, l_0 +2) \cup R( l_0 -1,l_0, l_0+2, l_0 +2)$ is not $G(l_0-1, l_0)$ corresponding to $ S_2$ in Fig.4.6 but, rather, the saddle $S_3$ in Fig.4.6; in other words starting from the birectangle $ R(l_0,l_0)$ it is no more convenient to continue the contraction along frame shapes but, on the contrary, the internal rectangle starts its independent shrinking keeping fixed the external one. It appears clear that if $h < 2 \l$ then the shrinking and disappearing of the internal two by two rectangle is just a down hill path where the number of internal plus spins decreases monotonically to zero. If we were considering a general initial condition corresponding to the above case 1 namely the contraction of a subcritical frame for $h > 2\l$, then we would have had $ l_0 >2$ and the shrinking and disappearence of the internal rectangle would have followed a sequence of squared or almost squared rectangular shape exactly like in the case of the standard Ising model. \par In the following we will consider birectangles $R(L_1,L_2;M_1,M_2)$ (see $\equ (3.3)$) also for $L_1,L_2=0,1$. \par Then the third part for $h < 2 \l$ is just the downhill path: $$ \bar {\cal T}_3 = S_3^*, R(1,2 ; 4,4),R(1,1 ; 4,4),R(0,0 ; 4,4) \Eq (7.9) $$ where by $S_3^*$ we denote the saddle depicted in Fig.4.6 when the external rectangle is a $ 4 \times 4$ square and the internal cluster is a ``triangle made by 3 sites".\par Finally the fourth part is just an Ising-like contraction of the remaining $ 4 \times 4$ rectangle of zeroes. We will observe first a sequence of permanece sets (corresponding to stable rectangles) and saddles and finally the downhill path describing the disappearence of the last $ 2 \times 2$ stable rectangle. \par We have: $$ \bar {\cal T}_4 = \tilde S _1 , \tilde R _1, \tilde S _2 , \tilde R _2, \tilde S _3 , \tilde R _3, \tilde S _4 , \tilde R _4, \tilde \o \Eq (7.10) $$ where $ \tilde R _1 = R(0,0 ; 4,3) \cup R(0,0 ; 3,4)$, $\tilde R _2 = R(0,0 ; 3,3)$, $\tilde R _3 = R(0,0 ; 3,2)\cup R(0,0 ; 2,3)$, $\tilde R _4 = R(0,0 ; 2,2)$; the downhill path $\tilde \o$ is given by $$ \tilde \o := \tilde S_5, \tilde R_5, \tilde R_6,\menouno\;\; , \Eq (7.11) $$ with $\tilde R_5 = R(0,0 ; 1,2)\cup R(0,0 ; 2,1)$, $\tilde R_6 = R(0,0 ; 1,1)$; the saddles $ \tilde S_i $, $i= 1, \dots , 5$ are obtained from the rectangles in $ \tilde R _i$ by adding a unit square protuberance to one of its longer sides.\par This concludes the definition of $\bar {\cal T}$ for $ h < 2\l, \d \; < \; { h +\l \over 2h}$.\par In the case $ h<2\l$, $\d \; > \; { h +\l \over 2h}$ the definition of $\bar {\cal T}$ is almost identical; we only have to modify a little bit at the very beginning the definition of $\bar {\cal T}_1$ by eliminating its first permanence set.\par Indeed we know from Section 5 that now $ H( S( l^*-1, l^*) ) > H( S( l^*, l^*) )$ so that the protocritical saddle is, in this case, ${\cal P}_{1,b } = S(l^* - 1, l^*)$. Now the configuration $\bar {\cal P}_1$ obtained from ${\cal P}_1$ by erasing the unit square protuberance is the subcritical rectangle $\bar {\cal P}_1 = R ( l^*-1, l^*)$; again this belongs to case 1. Then the first permanence set is now $A_{l^*-1,l^*-1}$ and we have $$ \bar {\cal T}_1 = A_{l^*-1,l^*-1}, S(l^*-1,l^*-1),A_{l^*-2,l^*-1}\dots S(\widetilde L,\widetilde L+1), A_{\widetilde L,\widetilde L}, S(\widetilde L,\widetilde L)\;\; . \Eq (7.12)$$ The other segments of the tube $\bar {\cal T}_i , \; i=2,3,4$ are defined exactly as before.\par The last case that we have still to analyze to construct $\bar {\cal T}$ is $h > 2 \l$. In this case the protocritical saddle is ${\cal P} = {\cal P}_2$ and the tube $\bar {\cal T} $ is just an Ising-like contraction along squared or almost squared rectangular clusters of zeroes in a sea of minuses. \par Now the configurations obtained by erasing the unit square protuberance, containing a unique subcritical rectangle of zeroes in a sea of minuses is given by: $$ \bar {\cal P}_2 = \bar R(M^*-1,M^*)\;\; ; $$ notice that $\bar {\cal P}_2$ is included in the case 2 above. \par We observe that the appearence of a single plus spin will induce the overcoming of an energy barrier greater or equal to $4J -(h + \l)$. It is very easy to see that we can proceed in the construction of the set of the standard cascades emerging from $\bar R(M^*-1,M^*)$ without be forced to overcome a barrier larger then $2J$ so that certainly in all these standard cascades, for our choice of the parameters, we will never see a single plus spin appearing. Indeed one easily convince himself that the sequence of minimax problems to be solved are the exact analogue of the ones arising in the analysis of a subcritical contraction for a standard Ising model. We refer to [S1], [KO1] for more details. For completeness in the following we summarize the results using our notation. \par The first permanence set is $\bar B(R(M^*-1,M^*))$.\par Let us define the following sequences of couples of integers: $$ (l_1,m_1), (l_2,m_2), \dots , (l_N,m_N) \;\;\; N = 2M^* -2 $$ $$ (l_1,m_1) = (M^* - 1, M^*), \;\;\; (l_N,m_N) = (1,1); \;\;\;\; |l_i-m_i| \leq 1 \; : \; \; m_i=l_i \;\; \hbox {or }\;\;\;m_i=l_i+1 $$ $$ \hbox {if }(l_i,m_i) = (l,l+1) \;\;\; \hbox {then } (l_{i+1},m_{i+1}) = (l,l) $$ $$ \hbox {if }(l_i,m_i) = (l,l)\;\;\; \hbox {then } (l_{i+1},m_{i+1}) = (l-1,l)\;\; . $$ Given $(l,m)$ as before: for $|l-m| \leq 1, \; 1 \leq m \leq M^*-1$ we denote by $ \bar S(l,m) $ the saddle obtained from $\bar R(l,m)$ by adding a unit square protuberance (with a zero spin inside) to one of its longest sides. \par We have: $$ \bar {\cal T} = \bar B ( \bar R(l_1,m_1)), \bar S (l_2,m_2), \bar B (\bar R(l_2,m_2) ), \dots , \bar S(l_N,m_N), \bar R(l_N,m_N), \menouno \;\; . $$ This concludes the definition of $\bar {\cal T}$.\par\bigskip Let us now pass to the definition of the descent part ${\cal T}_d$ of the tube ${\cal T}$. \par We start from the case $ h < 2\l, \d > { h+\l \over 2h}$ .\par It is immediately seen that by adding to ${\cal P}_1 = {\cal P}_{1,b} \equiv S(l^* -1,l^*)$ a unit square protuberance to form a stable protuberance of length 2 we get a configuration $\h_0$ included in case 3. \par We distinguish in ${\cal T}_d$ two parts: ${\cal T}_{d,1}$ and ${\cal T}_{d,2}$. \par For $l^* -1 \leq l < \widetilde M -2$ we denote by $ \bar A_{l-1,l}$ the cycle given by the maximal connected set of configurations containing $ C({l-1, l})$ with energy less than $ H(S(l,l))$. We easily verify that $F(\bar A_{l-1,l}) = C({l-1, l})$, ${\cal S}(\bar A_{l-1,l}) = S(l-1, l)$. \par For $l^* \leq l < \widetilde M -2$ we denote by $ \bar A_{l,l}$ the cycle given by the maximal connected set of configurations containing $ C({l, l})$ with energy less than $ H(S(l,l+1))$. We easily get that $F(\bar A_{l,l}) = C({l, l})$, ${\cal S}(\bar A_{l,l}) = S(l, l+1)$. \par For $l^* -1 \leq l < \widetilde M -2$ we denote by $ \O _{l-1,l}$ the set of downhill paths starting from $S(l-1, l)$ and ending in $\bar A _{l-1, l} $. Similarly, for $l^* \leq l < \widetilde M -2$ we denote by $ \O _{l,l}$ the set of downhill paths starting from $S(l, l)$ and ending in $\bar A _{l, l} $. We set $$ {\cal T}_{d,1} = \h_0, \bar A _{l^*-1, l^*}, S(l^*,l^*), \O _{l^*,l^*}, \bar A _{l^*, l^*},S(l^*,l^*+1),\O _{l^*,l`^*+1}, \dots, S({\widetilde M}-2,{\widetilde M}-2)\;\; . $$ As it has been shown in Section 4 for $l \geq \widetilde M -2$ the growth is typically symmetric in the sense that the probability of growth in the directions parallel or orthogonal to the shortest side of our supercritical frame are logarithmically equivalent for large $\b$. Moreover it follows from the analysis developed in Section 4 that for $l \geq \widetilde M -2$ the set ${\cal D}$ defined in $\equ (4.20)$ do not play any particular role and the permanence sets are cycles given by the strict basins of attraction of frames $C(l_1,l_2)$ or birectangles $R(l_1,l_2)$ . The second part ${\cal T}_{d,2}$ of ${\cal T}_d$ will describe the supercritical growth starting from $l= \widetilde M -2$. To construct ${\cal T}_{d,2}$ we need some more geometrical defininitions.\par For a given frame $C(l_1,l_2)$, we use the notation $C(l,m)$ to make explicit the shorter and longer sides $l$ and $m$, respectively.\par We denote by $G_>(l,m)$, $G_<(l,m)$, respectively, the saddle configurations containing a unique droplets obtained by attaching a unit square protuberance (with a zero spin inside) to a longer or shorter external side of $C(l,m)$.\par We denote by $R_>(l,m)$, $R_<(l,m)$, respectively, the birectangles obtained from $G_>(l,m)$, $G_<(l,m)$ by extending the unit square protuberance to an entire side.\par We denote by $S_>(l,m)$, $S_<(l,m)$, respectively, the saddle configurations containing a unique droplet obtained from $R_>(l,m)$, $R_<(l,m)$ by attaching a unit square protuberance (with a plus spin inside) to the internal free side.\par We denote by $\O_>(l,m)$, $\O_<(l,m)$, respectively, the set of all the downhill paths emerging from $S_>(l,m)$, $S_<(l,m)$ and ending in $\bar B(C(l+1,m))$, $\bar B(C(l,m+1))$; finally we denote by $\hat\O _>(l,m)$, $\hat\O _<(l,m)$ the set of all downhill paths emerging from $G_>(l,m)$, $G_<(l,m)$ and ending in $\bar B(R_>(l,m))$, $\bar B(R_<(l,m))$.\par Given $(l,m)$, we denote by $\G _>(l,m)$ the sequence: $\bar B(C(l,m))$, $G_>(l,m)$, $\hat\O _>(l,m)$, $\bar B(R_>(l,m))$, $S_>(l,m)$, $\O_>(l,m)$, $\bar B(C(l,m+1))$. Similarly we denote by $\G _<(l,m)$ the sequence: $\bar B(C(l,m))$, $G_<(l,m)$, $\hat\O _<(l,m)$, $\bar B(R_<(l,m))$, $S_<(l,m)$, $\O_<(l,m)$, $\bar B(C(l+1,m))$. \par A sequence $(l_i,m_i)_{i=1,2,\dots}$ with $l_i\leq m_i$ is called {\it regularly increasing} if: \par $l_1=m_1= \widetilde M-2$ and, for any $i=1,2, \dots$, either $(l_{i+1},m_{i+1}) \equiv (l_i,m_i)^> := (l_{i}+1,m_{i})$ or $(l_{i+1},m_{i+1}) \equiv (l_i,m_i)^< := (l_{i},m_{i}+1)$.\par If $l_i = m_i = L \; \equiv $ the side of our torus $\L$, we set $l_{i+1} = m_{i+1} = L$.\par Let ${\cal L}$ be the set of all regularly increasing sequences. For any $(l_i,m_i)_{i=1,2,\dots} \in {\cal L}$ we define: $\d (l_i,m_i) := \; >$ if $ (l_{i+1},m_{i+1}) \equiv (l_i,m_i)^>$ and $\d (l_i,m_i) := \; <$ if $ (l_{i+1},m_{i+1}) \equiv (l_i,m_i)^<$. \par >From the arguments developed in Section 4 it easily follows that the second part of ${\cal T}_{d}$ is given by: $$ {\cal T}_{d,2} = \cup _{(l_i,m_i)_{i=1,2,\dots} \in {\cal L}} \G_{\d(l_1,m_1)}(l_1,m_1) \cup \G_{\d(l_2,m_2)}(l_2,m_2) \cup \dots,\G_{\d(l_i,m_i)}(l_i,m_i), \dots\;\; . $$ This concludes the construction of ${\cal T}_{d}$ for the case $ h < 2\l, \d > { h+\l \over 2h}$ .\par The case $ h < 2\l, \d < { h+\l \over 2h}$ requires only minor changes: the only difference is that now we have to start a step further. Indeed it is immediately seen that by adding to ${\cal P}_1 = {\cal P}_{1,a} \equiv S(l^* ,l^*)$ a unit square protuberance to form a stable protuberance of length 2 we get a configuration $\h_0$ included in case 4. We have $$ {\cal T}_{d,1} = \h_0, \bar A _{l^*, l^*},S(l^*,l^*+1),\O _{l^*,l`^*+1}, \dots, S(M^*-2,M^*-2)\;\; . $$ The rest is identical.\par For the case $ h > 2\l$ we have exactly the same behaviour as in the Ising model namely we pass to consider an initial condition like in the case 5. Then we have a symmetric growth along a sequence of supercritical growing rectangles of zeroes in a sea of minuses up to the configuration $\zero$. Subsequently we have again an Ising-like nucleation of a protocritical droplet of pluses in the sea of zeroes (an $L^*\times (L^* -1)$ rectangle with a unit square protuberance attached to one of its longer sides) up to the configuration $\piuuno$. This last case has been already analyzed in detail (see, for instance [NS1], [S1]). We leave the details to the reader.\par One can easily convince himself that this indeed concludes the construction of the set of all standard cascades emerging from any of the above specified five type of initial conditions for any value of the parameters (not only for the subcases that we have explicitely treated).\par We can now state our main result on the tube of typical trajectories during the first excursion between $\menouno$ and $\piuuno$.\par Let ${\cal T} := {\cal T}_u \cup {\cal P} \cup {\cal T}_d$ with ${\cal T}_u$ given by the time reversal of the set of standard cascades in $\bar A \subset {\cal G} $ emerging from $ \bar {\cal P}$: ${\cal T}_u := {\cal R} \bar {\cal T} $ (the time reversal operator acts on paths in this way: for $\o = (x_1, x_2, \dots x_{N-1}, x_N) :\;\;{\cal R} \o = (x_N, x_{N-1}, \dots x_2, x_1)$); ${\cal T}_d$ given by the set of standard cascades in $\tilde A \subset {\cal G}^c$ emerging from $ \tilde {\cal P}$. Let $\bar {\cal P}_1$ be either $\bar {\cal P}_1$ or $\bar {\cal P}_2$ according to the values of the parameters $J,h,\m$; let $\tilde {\cal P}_1$ be either $\tilde {\cal P}_1$ or $\tilde {\cal P}_2$ according to the values of the parameters $J,h,\m$.\par %%%%%%%%%%%%%%%%%Theorem 2 \vskip 0.35 truecm \noindent {\bf Theorem 2.}\par\noindent Consider the dynamical Blume-Capel model described by the Markov chain with transition probabilities given in $\equ (2.5)$ of Section 2. For any choice of the parameters $J,h,\m$ compatible with $\equ (3.14)$.\par\noindent $i)$ $$ \lim_{\b \to \infty} P_{\menouno} ( \s_t \in {\cal T} \; \forall \; t \in [\bar \t _{\menouno}, \t _{\piuuno}] ) \; = \; 1\;\; . $$ The history of the process in ${\cal T}$ is described in the following way: \par \noindent consider an initial configuration $\h_0 $ corresponding to one of the following five cases:\par 1. $A = \bar A$: $\h_0 \in \bar A\cap [R(l,l)\cup R(l,l+1)]$ for some $l\ge \widetilde L$. \par 2. $A = \bar A$: $\h_0 \in \bar R(M^*,M^*-1)$.\par 3. $A = \tilde A$, $0 < \l < h <2\l$ and $\d > { h+\l \over 2h}$: $\h_0 \in \bar B (C(l^*-1,l^*))$.\par 4. $A = \tilde A$, $0 < \l < h <2\l$ and $\d < { h+\l \over 2h}$: $\h_0 \in \bar B (C(l^*,l^*) )$.\par 5. $A = \tilde A$: $\h_0 \in \bar B (\bar R(M^*,M^*))$. \par Then, considering for any such $A , \h_0$ the set of all standard cascades emerging from $\h_0$ and falling into $F(A)$ we have \par \noindent $ii)$ $$ \exists \; \d >0 \;\; \hbox {such that}\;\; \lim _{ \b \to \infty} P_{\h_0} (\t _{ F(A)} < \exp ( \b [ H(\h_1) - H(F(A)) - \d] )=1\;\; , $$ \par \noindent $iii)$ $$ \lim _{ \b \to \infty} P_{\h_0} (\forall \; t \leq \t _{ F(A)} \;:\; x_t \; \in \;{\cal T} ( \h_0, \o_1, \h_1,\o_2, \dots , \h_{M-1}, \o_M) $$ $$ \hbox { for some standard cascade }\; \h_0, \o_1, \h_1,\o_2, \dots , \h_{M-1}, \o_M ) \; =\; 1\;\; , $$ \par \noindent $iv)$ moreover, with probability $\to \; 1$ as $\b \to \infty$, there exists a sequence $\h_0, \o_1, \h_1,\o_2,$ $ \dots , \h_{M-1}, \o_M $ such that our process starting at $t=0$ from $\h_0$, between $t=0$ and $t= \t _{ F(A)}$, after having followed the initial downhill path $\o_1$, visits, sequentially, the sets $A_1, A_2,\dots , A_{M-1}$ exiting from $A_j$ through $\h_j$ and then following the path $\o_{j+1}$ before entering $A_{j+1}$.\par For every $\e >0$ with probability tending to one as $\b \to \infty$ the process spends inside each $A_j$ a time $T_j (\e)$ : $ \exp ( \b [ H(\h_j) - H(F(A_j)) - \e] ) \; < \; T_j (\e) \;< \; \exp ( \b [ H(\h_j) - H(F(A_j)) + \e] ) $ and before exiting from $A_j$ it visits each point in $A_j$ at least $\exp \b \e$ times . %%%%%%%%%%%%%%%%%%%End of Theorem 2. \par\noindent {\it Proof.} \vskip 0.5 truecm \par We easily get that $$ \lim _{\b \to \infty} P_{ {\cal P}_1}(\s_1 \in \bar {\cal P}_1| \s_1 \in {\cal G}) = 1 \;\; .\Eq (7.20) $$ Indeed \equ (7.20) follows from the fact that there is only one first possible step in any downhill path from ${\cal P}_1$ to ${\cal G}$: it corresponds to erasing the unit square protuberance to get $\bar {\cal P}_1$. \par On the other hand we have: $$ \lim _{\b \to \infty} P_{ {\cal P}_{1,b}}(\s_1=\h_0 | \s_1 \in {\cal G}^c) = 1\;\; .\Eq (7.21) $$ The proof is an immediate consequence of Theorem 1, \equ (7.20), \equ (7.21), Theorem 1 in [OS1], and the results of [S1]. $\square$ %%Pagina con la figure sulle selle. \vfill\eject %\special{psfile="global.ps" voffset=-650 hoffset=-110} \par $\phantom .$ \vfill\eject \numsec=8\numfor=1 \vskip 1cm {\bf Section 8. Conclusions.} \par We have described the metastable behavior of a dynamical Blume-Capel model. Our updating rule is given by the classical Metropolis algorithm but it is clear that our results extend to a wide class of single-spin-flip reversible dynamics.\par Our results refer to the asymptotic regime of small but fixed magnetic field $h$ and chemical potential $\l$, large but fixed volume $\L$ and very large inverse temperature $\b$.\par We take mainly the point of view of the so called {\it pathwise approach} to metastability aiming to describe the typical behaviour of the random trajectories of our stochastic dynamics rather than describing the evolution of the averages.\par Blume-Capel model exhibits the interesting feature of the presence of three possible phases. The equilibrium phase diagram is, consequently, very reach and interesting. The most important aspect from the point of view of the study of the dynamics of metastability is the presence, near the triple point, of two competing metastable phases. This means that, for instance, if one wants to describe the decay from the metastable $\menouno$ phase to the stable $\piuuno$ phase one has to take into account the presence of another metastable phase: $\zero$.\par We took as initial condition the state $\menouno$ and we analyzed the region of parameters $0<\l H(\zero) > H(\piuuno)\;\; . $$ In the region $$ {\rm I} := 0< h< \l$$ we have $$ H(\zero) > H(\menouno) > H(\piuuno) $$ and then it is reasonable to expect and not difficult to prove that in the decay from $\menouno$ to $\piuuno$ the state $\zero$ does not play any role. Indeed it is sufficient to exhibit a mechanism of transition from $\menouno$ to $\piuuno$ involving an energy barrier smaller than $H(\zero) - H(\menouno) $.\par This is very easy to achieve if the volume $\L$ is sufficiently large.\par In this paper we analyzed in detail the regions II and III which happen to be, in a sense, the most interesting ones. In the region IV one has the same local minima for the energy as in the regions II and III; they are sets of non-interacting plurirectangles; but now the comparison between the times $t_1,t_2,t_3,t_4$ introduced in $\equ (3.11)$ changes totally. The main difference w.r.t. the regions II, III is that now, in IV, we have: $$ M^* < L^*\;\; , $$ and so we cannot even consider a possible mechanism of nucleation along a sequence of frames. Indeed one has that a birectangle is supercritical if and only if the minimal external side is not smaller than $M^*$. Then, like in region III but in a much easier way, we can prove that the escape from $\menouno$ starts with an Ising-like nucleation of a protocritical droplet ${\cal P}_2$ leading to $\zero$. But now, contrary to the region III the typical time $T^{\menouno \to \zero}$ for going from $\menouno$ to $\zero$ is much shorter than the typical time $T^{\zero \to \piuuno }$ for going from $\zero$ to $\piuuno$ so that the asymptotics of the time $T^{\menouno \to \piuuno }$ of the transition from $\menouno$ to $\piuuno$ is dominated by $T^{\zero \to \piuuno }$. \par The situation in which a priori one could expect a competition between the two metastable phases would be at a first glance the union of the regions II, III, IV. By arguing more carefully with a heuristic analysis of the heights of the possible barriers between $\menouno$ and $\zero$ and between $\menouno$ and $\piuuno$ (given by the energy of formation of suitable critical droplets) one is led to expect that the two metastable phases corresponding to $\menouno$ and $\zero$ are in a sense really competing only around the half line $0< h= 2\l $ separating the regions II and III. This value $h= 2\l $ depends on the particular form of the Blume-Capel hamiltonian.\par The main result of the present paper consists in the rigorous proof of the above heuristics.\par >From mathematical point of view we had to solve some large deviation problems. This kind of problems would be extremely hard for a general non-reversible dynamics but their treatment is very much simplified by the reversibility property of the dynamics.\par In particular to get the result we had to solve the minimax problem of the determination of the global saddle between $\menouno$ and $\piuuno$. This is the really hard point of the work. We could handle the large deviation problems \`a la Freidlin-Wentzell arising in the study of some rare events in the framework of our low temperature Metropolis dynamics by taking advantage of a general approach to the study of typical trajectories, during the first exit from a non-completely attracted domain, recently developed in [OS1]. Nevertheless we still had to face the crucial model-dependent part consisting in solving some geometrically quite involved variational problems.\par In particular we had to exclude, as highly depressed in probability, any mechanism of transition based on coalescence and we had to single out, among many others, only very few possible mechanisms of nucleation. \par \midinsert \vskip 11 truecm\noindent %\special{psfile="fig8_1.ps" voffset=-300 hoffset=-100} \vskip 1 truecm \centerline {\smbfb Fig.8.1} \endinsert \par We were able to rigorously compute the lifetime of the metastable state, namely the tipical transition times $T_{\l,h}$, for different values of the parameters $\l,h$. It turns out that these transition times are given by $$ T_{\l,h} \sim \exp (\b \G _{\l,h})$$ where the ``activation energy", for very small values of $\l,h$ has the following expression $$ \G _{\l,h} \sim {8J^2 \over h} \;\;\;\; \hbox {for} \;\;\;\; 0< \l \G^{\zero \to \piuuno}_{\l,h} $ and this is the reason for \equ (8.2); but in Region IV we have the opposite $\G^{\menouno \to \zero}_{\l,h} < \G^{\zero \to \piuuno}_{\l,h}$ so that we get : $$ \G _{\l,h} \sim {4J^2 \over h+\l} \;\;\;\; \hbox {for} \;\;\;\;0< -\l \; 0\} \;\; ;\eqno (A.6) $$ let $V=V(A)$ be the analogue of $R$ outside $A$: $$ V(A) := \{ y \not \in A \;\; {\rm such \; that}\;\; \exists z \; \in \; {\cal S}(A)\;\; {\rm with} \;\; P(x,y) \; > \; 0\} \;\; .\eqno (A.7) $$ We set: $${\cal H} := R(A) \cup V(A)\;\; .\eqno (A.8)$$ %%%%%%%%%%%%%%%%%Proposition A.1 \vskip 0.35 truecm \noindent {\bf Proposition A.1.}\par\noindent Consider a transient cycle $A$. Given $\e >0$ let $$T(\e) := \exp \b [H({\cal S}(A))-H(F(A)) + \e]\;\; .\eqno (A.9)$$ Then, for every $\e>0, \; x \in A$, $$ \lim_{\b \to \infty} P_x( \t_{ (A\cup \partial A)^c} \; > \; T(\e) )\; = \; 0 \;\; ,\eqno (A.10) $$ and $$ \lim_{\b \to \infty} P_x(X_{ \t_{ (A\cup \partial A)^c}} \in \; V(A) )\; = \; 1 \;\; .\eqno (A.11) $$ %%%%%%%%%%%%%%%%%%%End of Proposition A.1. \par\noindent {\it Proof.} \vskip 0.5 truecm \par >From Hypothesis M and the definition of ${\cal S}(A)$ we know that there exists a positive constant $c > 0$, independent of $\b$, such that $$ \inf _{ x\in {\cal S}(A),y\in {\cal H}} P(x,y) \; > \; c,\;\;\;\;\;\;\;\; \lim _{\b \to \infty}\sup _{ x\in {\cal S}(A),y \not \in {\cal H}} P(x,y) \; = \; 0 \;\; .\eqno (A.12) $$ We define, now, the sequence $\t_i$ of stopping times corresponding to subsequent passages of our chain $X_t$ in $\partial A$: $$\eqalign{ \t_o :=&\inf \{ t\geq 0 : X_t \in \partial A\}\cr \sigma_1 :=&\inf \{ t > \t_o : X_t \not\in \partial A\}\cr} \;\; ,\eqno (A.13)$$ and for $j=1,2,\dots$: $$\eqalign{ \t_{j} :=&\inf \{ t > \s_{j} : X_t \in \partial A\}\cr \s_{j} :=&\inf \{ t > \t_{j-1} : X_t \not\in \partial A\}\cr} \;\; .\eqno (A.14) $$ We set, for $j=1,2,\dots$: $$ {\cal I}_j = [\t_{j-1} +1, \t_j]\; ,\;\;\;\;\;\; T_j:=|{\cal I}_j|= \t_j - \t_{j-1} -1 \;\; . \eqno (A.15) $$ Suppose that $ X_{\t_{j-1} +1} \; \in \; A$; let $$\s^*_j := \min \{ t\; >\; \t_j \; : \; X_t \neq X_{\t_j}\}\; ; \eqno (A.16) $$ we say that the interval ${\cal I}_j$ is {\it good} if the following conditions are satisfied: $$\eqalign{ T_j &< T(\e) \cr X_{\t_j} &\in {\cal S}(A)\cr X_{\s^*_j} &\in{\cal H}\cr}\;\; . $$ Let $$ j^* := \min \{j \; : \; T_j \hbox { is not good}\}\;\; .$$ Given the integer $N$ we want to estimate, for every $x \in A$, the probability $P_x ( \t_{V(A)} > N T(\e) )$.\par We write: $$ P_x ( \t_{V(A)}\ > N T(\e) )= P_x ( \t_{V(A)} > N T(\e)\; ; \; j^* > N )+P_x ( \t_{V(A)} > N T(\e)\; ; \; j^* \leq N) \eqno (A.17) $$ Let us consider the first event in the decomposition given in $(A.17)$: $\{ \t_{V(A)} > N T(\e) ; \; j^* > N \}$.\par We have: $$ P_x ( \t_{V(A)}\; > \; N T(\e)\; ; \; j^* \; >\; N )\; \leq $$ $$ \leq P_x (X_{\t_1} \in {\cal S}(A);X_{\t_1+1} \not \in V(A), \dots , X_{\t_N} \in {\cal S}(A);X_{\t_N+1} \not \in V(A)) \leq $$ $$ \leq ( 1-c)^N\;\; . \eqno (A.18) $$ Now, from Proposition 3.7 of [OS1], $(A.12)$ and the stationarity of our Markov chain we have : $$ P_x ( {\cal I}_j \hbox { is not good} ) \; \leq \d (\b) \eqno (A.19) $$ with $$ \lim _{\b \to \infty} \d (\b) \; = \; 0\;\; . \eqno (A.20) $$ >From $(A.20)$ we get: $$ P_x ( \t_{V(A)}\; > \; N T(\e)\; ; \; j^* \; \leq\; N)\; \leq \; P_x ( j^* \; \leq\; N)\; \leq \; \d(\b) N\;\; . \eqno (A.21) $$ To conclude the proof it suffices to choose : $$ N \;= \;N(\b)\; = \;1/\d(\b). $$ \vfill\eject \numsec=10\numfor=1 \centerline {\bf Acknowledgements.} \vskip 1 truecm \par\noindent We acknowledge R. 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