%% !!!!! to be typesetted twice !!!!!
%%%%%%%%%%%%%%%%macros start here%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%begin of Gallavotti's macros%%%%%%%%
\newcount\driver \newcount\mgnf \newcount\tipi
\newskip\ttglue
%%cm completo
\def\TIPITOT{
\font\dodicirm=cmr12
\font\dodicii=cmmi12
\font\dodicisy=cmsy10 scaled\magstep1
\font\dodiciex=cmex10 scaled\magstep1
\font\dodiciit=cmti12
\font\dodicitt=cmtt12
\font\dodicibf=cmbx12 scaled\magstep1
\font\dodicisl=cmsl12
\font\ninerm=cmr9
\font\ninesy=cmsy9
\font\eightrm=cmr8
\font\eighti=cmmi8
\font\eightsy=cmsy8
\font\eightbf=cmbx8
\font\eighttt=cmtt8
\font\eightsl=cmsl8
\font\eightit=cmti8
\font\seirm=cmr6
\font\seibf=cmbx6
\font\seii=cmmi6
\font\seisy=cmsy6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\font\dodicitruecmr=cmr10 scaled\magstep1
\font\dodicitruecmsy=cmsy10 scaled\magstep1
\font\tentruecmr=cmr10
\font\tentruecmsy=cmsy10
\font\eighttruecmr=cmr8
\font\eighttruecmsy=cmsy8
\font\seventruecmr=cmr7
\font\seventruecmsy=cmsy7
\font\seitruecmr=cmr6
\font\seitruecmsy=cmsy6
\font\fivetruecmr=cmr5
\font\fivetruecmsy=cmsy5
%%%% definizioni per 10pt %%%%%%%%
\textfont\truecmr=\tentruecmr
\scriptfont\truecmr=\seventruecmr
\scriptscriptfont\truecmr=\fivetruecmr
\textfont\truecmsy=\tentruecmsy
\scriptfont\truecmsy=\seventruecmsy
\scriptscriptfont\truecmr=\fivetruecmr
\scriptscriptfont\truecmsy=\fivetruecmsy
%%%%% cambio grandezza %%%%%%
\def \ottopunti{\def\rm{\fam0\eightrm}% switch to 8-point type
\textfont0=\eightrm \scriptfont0=\seirm \scriptscriptfont0=\fiverm
\textfont1=\eighti \scriptfont1=\seii \scriptscriptfont1=\fivei
\textfont2=\eightsy \scriptfont2=\seisy \scriptscriptfont2=\fivesy
\textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
\textfont\itfam=\eightit \def\it{\fam\itfam\eightit}%
\textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}%
\textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}%
\textfont\bffam=\eightbf \scriptfont\bffam=\seibf
\scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}%
\tt \ttglue=.5em plus.25em minus.15em
\setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}%
\normalbaselineskip=9pt
\let\sc=\seirm \let\big=\eightbig \normalbaselines\rm
\textfont\truecmr=\eighttruecmr
\scriptfont\truecmr=\seitruecmr
\scriptscriptfont\truecmr=\fivetruecmr
\textfont\truecmsy=\eighttruecmsy
\scriptfont\truecmsy=\seitruecmsy
}\let\nota=\ottopunti}
\newfam\msbfam %per uso in \TIPITOT
\newfam\truecmr %per uso in \TIPITOT
\newfam\truecmsy %per uso in \TIPITOT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%cm ridotto
\def\TIPI{
\font\eightrm=cmr8
\font\eighti=cmmi8
\font\eightsy=cmsy8
\font\eightbf=cmbx8
\font\eighttt=cmtt8
\font\eightsl=cmsl8
\font\eightit=cmti8
\font\tentruecmr=cmr10
\font\tentruecmsy=cmsy10
\font\eighttruecmr=cmr8
\font\eighttruecmsy=cmsy8
\font\seitruecmr=cmr6
\textfont\truecmr=\tentruecmr
\textfont\truecmsy=\tentruecmsy
%%%%% cambio grandezza %%%%%%
\def \ottopunti{\def\rm{\fam0\eightrm}% switch to 8-point type
\textfont0=\eightrm
\textfont1=\eighti
\textfont2=\eightsy
\textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
\textfont\itfam=\eightit \def\it{\fam\itfam\eightit}%
\textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}%
\textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}%
\textfont\bffam=\eightbf
\def\bf{\fam\bffam\eightbf}%
\tt \ttglue=.5em plus.25em minus.15em
\setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}%
\normalbaselineskip=9pt
\let\sc=\seirm \let\big=\eightbig \normalbaselines\rm
\textfont\truecmr=\eighttruecmr
\scriptfont\truecmr=\seitruecmr
%\textfont\truecmsy=\eighttruecmsy
}\let\nota=\ottopunti}
%%am
\def\TIPIO{
\font\setterm=amr7 %\font\settei=ammi7
\font\settesy=amsy7 \font\settebf=ambx7 %\font\setteit=amit7
%%%%% cambiamenti di formato %%%
\def \settepunti{\def\rm{\fam0\setterm}% passaggio a tipi da 7-punti
\textfont0=\setterm %\textfont1=\settei
\textfont2=\settesy %\textfont3=\setteit
%\textfont\itfam=\setteit \def\it{\fam\itfam\setteit}
\textfont\bffam=\settebf \def\bf{\fam\bffam\settebf}
\normalbaselineskip=9pt\normalbaselines\rm
}\let\nota=\settepunti}
%%%%%%%%% GRAFICA
%
% Inizializza le macro postscript e il tipo di driver di stampa.
% Attualmente le istruzioni postscript vengono utilizzate solo se il driver
% e' DVILASER ( \driver=0 ), DVIPS ( \driver=1) o PSPRINT (\driver=2);
% qualunque altro valore di \driver produce un output in cui le figure
% contengono solo i caratteri inseriti con istruzioni TEX (vedi avanti).
%
%\ifnum\driver=0 \special{ps: plotfile ini.pst global} \fi
%\ifnum\driver=1 \special{header=ini.pst} \fi
\newdimen\xshift \newdimen\xwidth \newdimen\yshift
%
% inserisce una scatola contenente #3 in modo che l'angolo superiore sinistro
% occupi la posizione (#1,#2)
%
\def\ins#1#2#3{\vbox to0pt{\kern-#2 \hbox{\kern#1 #3}\vss}\nointerlineskip}
%
% Crea una scatola di dimensioni #1x#2 contenente il disegno descritto in
% #4.ps; in questo disegno si possono introdurre delle stringhe usando \ins
% e mettendo le istruzioni relative nell'argomento #3.
% Il file #4.ps contiene le istruzioni postscript, che devono essere scritte
% presupponendo che l'origine sia nell'angolo inferiore sinistro della
% scatola, mentre per il resto l'ambiente grafico e' quello standard.
% #5 deve essere della forma \eq("nome simbolico").
%
% Le istruzioni postscript possono essere inserite nel file che contiene
% l'istruzione \insertplot, racchiudendole fra le istruzioni \initfig{#4}
% e \endfig; inoltre ogni riga deve cominciare con "write13<" e deve finire
% con ">". In questo modo si crea il file #4.ps relativo alla figura.
%
\def\insertplot#1#2#3#4#5{\par%
\xwidth=#1 \xshift=\hsize \advance\xshift by-\xwidth \divide\xshift by 2%
\yshift=#2 \divide\yshift by 2%
\line{\hskip\xshift \vbox to #2{\vfil%
\ifnum\driver=0 #3
\special{ps::[local,begin] gsave currentpoint translate}%
\special{ps: plotfile #4.ps} \special{ps::[end]grestore}\fi
\ifnum\driver=1 #3 \special{psfile=#4.ps}\fi
\ifnum\driver=2 #3 \special{
\ifnum\mgnf=0 #4.ps 1. 1. scale \fi
\ifnum\mgnf=1 #4.ps 1.2 1.2 scale\fi} \special{ini.ps}
\fi }\hfill \raise\yshift\hbox{#5}}}
\def\initfig#1{%
\catcode`\%=12\catcode`\{=12\catcode`\}=12
\catcode`\<=1\catcode`\>=2
\openout13=#1.ps}
\def\endfig{%
\closeout13
\catcode`\%=14\catcode`\{=1
\catcode`\}=2\catcode`\<=12\catcode`\>=12}
%%%%%%%%%%%%%%%% 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\ps=\psi
\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
%%%%%%%%%%%%%%%%%%%%% 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;\,\the\time}
\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}
\def\footnormal{
\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm
\foglio\hss}
}
\def\footappendix{
\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.\veraformula)%
\SIA e,#1,(A.\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
\write16{#1 non e' (ancora) definito}%
\else\csname fu#1\endcsname\fi}
\def\equ(#1){\senondefinito{e#1}\eqv(#1)\else\csname e#1\endcsname\fi}
\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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%% 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\ii=\int \let\ig=\int \let\io=\infty
\let\dpr=\partial \def\V#1{\vec#1} \def\Dp{\V\dpr}
\def\oo{{\V\o}} \def\OO{{\V\O}} \def\uu{{\V\y}} \def\xxi{{\V \xi}}
\def\xx{{\V x}} \def\yy{{\V y}} \def\kk{{\V k}} \def\zz{{\V z}}
\def\rr{{\V r}} \def\pp{{\V p}}
\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\guida{\ ...\ } \def\Z{{\bf Z}}\def\R{{\bf R}}\def\tab{}\def\nonumber{}
\def\mbox{\hbox}\def\lis#1{{\overline#1}}\def\nn{{\V n}}
\def\Tr{{\rm Tr}\,}\def\EE{{\cal E}}
\def\Veff{{V_{\rm eff}}}\def\Pdy{{P(d\psi)}}\def\const{{\rm const}}
%\def\RR{{\cal R}}
\def\NN{{\cal N}}\def\ZZ#1{{1\over Z_{#1}}}
\def\OO{{\cal O}} \def\GG{{\cal G}} \def\LL{{\cal L}} \def\DD{{\cal D}}
\def\fra#1#2{{#1\over#2}}
\def\ap{{\it a priori\ }}
\def\rad#1{{\sqrt{#1}\,}}
\def\eg{{\it e.g.\ }}
\def\={{\equiv}}\def\ch{{\chi}}
\def\initfiat#1#2#3{
\mgnf=#1
\driver=#2
\tipi=#3
\ifnum\tipi=0\TIPIO \else\ifnum\tipi=1 \TIPI\else \TIPITOT\fi\fi
%\ifnum\driver=0 \special{ps: plotfile ini.pst global} \fi
%\ifnum\driver=1 \special{header=ini.pst} \fi
%%%%%%%%%%%%%%% FORMATO
\ifnum\mgnf=0
\magnification=\magstep0 \hoffset=0.cm
\voffset=-1truecm\hoffset=-.5truecm\hsize=16.5truecm \vsize=25.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
\fi
\ifnum\mgnf=1
\magnification=\magstep1
\hoffset=0.cm
\voffset=-1truecm
\hoffset=-.5truecm
\hsize=16.5truecm
\vsize=25truecm
\baselineskip=12pt
% 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
\fi
\setbox200\hbox{$\scriptscriptstyle \data $}
}
%%%%%%%%%%%end of Gallavotti's macros%%%%%%%%%
%%%%%%%%%%%inizialization%%%%%%%%%%%
%%%%%%put % in front of \BOZZA to remove labels on the left%%%%%%%%%%%
\initfiat {1}{1}{2}
%\BOZZA
%\input amssym.def%
%%%%%%%%amssym.def included here%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\expandafter\ifx\csname amssym.def\endcsname\relax \else\endinput\fi
%
% Store the catcode of the @ in the csname so that it can be restored later.
\expandafter\edef\csname amssym.def\endcsname{%
\catcode`\noexpand\@=\the\catcode`\@\space}
% Set the catcode to 11 for use in private control sequence names.
\catcode`\@=11
%
% Include all definitions related to the fonts msam, msbm and eufm, so that
% when this file is used by itself, the results with respect to those fonts
% are equivalent to what they would have been using AMS-TeX.
% Most symbols in fonts msam and msbm are defined using \newsymbol;
% however, a few symbols that replace composites defined in plain must be
% defined with \mathchardef.
\def\undefine#1{\let#1\undefined}
\def\newsymbol#1#2#3#4#5{\let\next@\relax
\ifnum#2=\@ne\let\next@\msafam@\else
\ifnum#2=\tw@\let\next@\msbfam@\fi\fi
\mathchardef#1="#3\next@#4#5}
\def\mathhexbox@#1#2#3{\relax
\ifmmode\mathpalette{}{\m@th\mathchar"#1#2#3}%
\else\leavevmode\hbox{$\m@th\mathchar"#1#2#3$}\fi}
\def\hexnumber@#1{\ifcase#1 0\or 1\or 2\or 3\or 4\or 5\or 6\or 7\or 8\or
9\or A\or B\or C\or D\or E\or F\fi}
\font\tenmsa=msam10
\font\sevenmsa=msam7
\font\fivemsa=msam5
\newfam\msafam
\textfont\msafam=\tenmsa
\scriptfont\msafam=\sevenmsa
\scriptscriptfont\msafam=\fivemsa
\edef\msafam@{\hexnumber@\msafam}
\mathchardef\dabar@"0\msafam@39
\def\dashrightarrow{\mathrel{\dabar@\dabar@\mathchar"0\msafam@4B}}
\def\dashleftarrow{\mathrel{\mathchar"0\msafam@4C\dabar@\dabar@}}
\let\dasharrow\dashrightarrow
\def\ulcorner{\delimiter"4\msafam@70\msafam@70 }
\def\urcorner{\delimiter"5\msafam@71\msafam@71 }
\def\llcorner{\delimiter"4\msafam@78\msafam@78 }
\def\lrcorner{\delimiter"5\msafam@79\msafam@79 }
\def\yen{{\mathhexbox@\msafam@55 }}
\def\checkmark{{\mathhexbox@\msafam@58 }}
\def\circledR{{\mathhexbox@\msafam@72 }}
\def\maltese{{\mathhexbox@\msafam@7A }}
\font\tenmsb=msbm10
\font\sevenmsb=msbm7
\font\fivemsb=msbm5
\newfam\msbfam
\textfont\msbfam=\tenmsb
\scriptfont\msbfam=\sevenmsb
\scriptscriptfont\msbfam=\fivemsb
\edef\msbfam@{\hexnumber@\msbfam}
\def\Bbb#1{{\fam\msbfam\relax#1}}
\def\widehat#1{\setbox\z@\hbox{$\m@th#1$}%
\ifdim\wd\z@>\tw@ em\mathaccent"0\msbfam@5B{#1}%
\else\mathaccent"0362{#1}\fi}
\def\widetilde#1{\setbox\z@\hbox{$\m@th#1$}%
\ifdim\wd\z@>\tw@ em\mathaccent"0\msbfam@5D{#1}%
\else\mathaccent"0365{#1}\fi}
\font\teneufm=eufm10
\font\seveneufm=eufm7
\font\fiveeufm=eufm5
\newfam\eufmfam
\textfont\eufmfam=\teneufm
\scriptfont\eufmfam=\seveneufm
\scriptscriptfont\eufmfam=\fiveeufm
\def\frak#1{{\fam\eufmfam\relax#1}}
\let\goth\frak
% Restore the catcode value for @ that was previously saved.
\csname amssym.def\endcsname
%
%%%%%%%%%%%%end of amssym.def%%%%%%%%
%
%%%%%%%%%%%%%%%%%%extra definitions already in yau's file%%%%%%%%
%
\def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt
\hbox{\vrule width.#2pt height#1pt \kern#1pt
\vrule width.#2pt}\hrule height.#2pt}}}}
\def\qed{ $\mathchoice\sqr64\sqr64\sqr{2.1}3\sqr{1.5}3$}
\def\ZZ{Z\!\!\!Z\,}
\def\RR{R\!\!\!\!\!I\,\,}
\def \II{\ \hbox{I}\!\!\!\hbox{I}\,}
\def\11{\hbox{l}\!\!\!1\,}
\def\QIF{\quad\hbox{ if }\quad}
\font\tenib=cmmib10
\newfam\mitbfam
\textfont\mitbfam=\tenib
\scriptfont\mitbfam=\seveni
\scriptscriptfont\mitbfam=\fivei
\def\mitb{\fam\mitbfam}
\def\balpha{{\mitb\mathchar"710B}}
\def\bbeta{{\mitb\mathchar"710C}}
\def\bgamma{{\mitb\mathchar"710D}}
\def\bdelta{{\mitb\mathchar"710E}}
\def\bepsilon{{\mitb\mathchar"710F}}
\def\bzeta{{\mitb\mathchar"7110}}
\def\boeta{{\mitb\mathchar"7111}} %bold eta
%above ceta because bold eta should not be beta
\def\btheta{{\mitb\mathchar"7112}}
\def\biota{{\mitb\mathchar"7113}}
\def\bkappa{{\mitb\mathchar"7114}}
\def\blambda{{\mitb\mathchar"7115}}
\def\bmu{{\mitb\mathchar"7116}}
\def\bnu{{\mitb\mathchar"7117}}
\def\bxi{{\mitb\mathchar"7118}}
%\def\bomicron{{\mitb\mathchar"7122}}
%above, not omicron, sort of script e lower case
\def\bpi{{\mitb\mathchar"7119}}
\def\brho{{\mitb\mathchar"701A}}
\def\bsigma{{\mitb\mathchar"701B}}
\def\btau{{\mitb\mathchar"701C}}
\def\bupsilon{{\mitb\mathchar"701D}}
\def\bchi{{\mitb\mathchar"701F}}
\def\bpsi{{\mitb\mathchar"7120}}
\def\bomega{{\mitb\mathchar"7121}}
%\baselineskip7mm
\def\and{ \hbox{ and } }
\def\la {\big\langle}
\def\ra {\big\rangle}
\def\pa{\parallel}
\def\pt{\partial}
\def\l{\lambda}
\def\L{\Lambda}
\def\e{\varepsilon}
\def\a{\alpha}
\def\be{\beta}
\def\c{\hbox{const.}}
\def\d{\delta}
\def\g{\gamma}
\def\o{\omega}
\def\om{\omega}
\def\O{\Omega}
\def\n{\nabla}
\def\s{\sigma}
\def\t{\theta}
\def\newpage{\vfill\eject}
\def\Cal{\cal}
\def\to{\rightarrow}
\def\if {\hbox { if } }
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\centerline {\dodicibf On the validity of the van der Waals theory}
\vskip.3cm
\centerline {\dodicibf in Ising systems with long
range interactions}
\vskip1cm
\centerline{
P. Butt\`a $^1$,
\hskip.2cm
I. Merola $^1$
\hskip.1cm and \hskip.1cm
E. Presutti \footnote{$^1$}{\eightrm Dipartimento di Matematica,
Universit\`a di Roma Tor Vergata, Via della Ricerca Scientifica,
00133 Roma, Italy} }
\footnote{}{\eightrm This research has been partially
supported by CNR, (GNFM and INFM) and by grant CEE CHRX-CT93-0411}
\vskip.5cm
\centerline {Universit\`a di Roma Tor Vergata}
\vskip.5cm
{\bf Abstract.} We consider an Ising system
in $d \ge 2$ dimensions with ferromagnetic
spin-spin
interactions $-J_\g(x,y)\s(x)\s(y)$, $x$, $y \in \Bbb Z^d$, where
$J_\g(x,y)$ scales like a Kac potential. We prove that when the
temperature is below the mean field critical value,
for any $\g$ small enough (i.e. when the range of the interaction
is long but finite), there are only two pure homogeneous phases,
as stated by the van der Waals theory. The proof follows that in
[\rcite{GM}] on the translationally invariant states at low
temperatures for nearest neighbor interactions, supplemented by
a ``relative
uniqueness criterion for Gibbs fields" which yields
uniqueness in a restricted ensemble of measures, in a
context where there is a phase transition. This
criterion is derived by introducing special couplings as in
[\rcite {BM}] which reduce the proof of relative
uniqueness to the absence of percolation of ``bad events".
\vskip1cm
{\it Keywords}: {\sl Kac potentials, Translationally invariant Gibbs states,
Couplings.}
\vskip1truecm
\centerline{\bf 1. Introduction.}
\vskip.5cm
\numsec= 1
\numfor= 1
To prove the validity of the van der Waals theory in an equilibrium
Statistical Mechanics setting, Kac, Uhlenbeck and Hemmer, [\rcite {KUH}],
have introduced interactions that
depend on a scaling parameter $\g >0$
studying the limit $\g \to 0^+$ where the range
of the interaction becomes infinite.
This program has been carried out by Kac, Uhlenbeck and Hemmer
in some particular models and then by Lebowitz and
Penrose in a more general class of systems, [\rcite {LP}].
These results prove that
the phase diagram (of the free energy versus
the magnetization at fixed temperature)
converges for any temperature
to the phase diagram of the
van der Waals theory. Notice however that this
does not imply that a
phase transition in the limit theory
occurs also before the limit, when the interaction
range, maybe very long, is still finite. Strangely enough
such a basic question (whose answer is obviously in the negative
for $d=1$ dimensions) has not been investigated
in $d\ge 2$ till recently
when in [\rcite {CP}] it has been proved that for
ferromagnetic Ising systems with ``pure" Kac potentials (those defined
in the next
section) and in $d\ge 2$ dimensions, the true
inverse critical temperature $\beta_{c}(\g)$ converges, as $\g\to 0^+$,
to its van der Waals value (that in our setup is equal to 1).
This result has then been extended in [\rcite {BP}] to
non trivial ``reference interactions" while in [\rcite {BZ}]
an upper bound on
$\beta_{c}(\g)$ is derived for pure Kac potentials while a lower
bound in $d=2$ is proved in [\rcite{CMP}]. We
complete here the proof of the validity
of the van der Waals theory for pure Kac potentials
by showing that not only the
critical temperature but also the whole phase diagram is
qualitatively the same at $\g>0$ (and small) as in the
van der Waals theory. Namely for any $\beta>1$ (i.e. below the van
der Waals critical temperature) there is $\g_\beta>0$ so that for
any $\g\le \g_\beta$ there are only two
pure homogeneous phases as in the van der Waals theory.
The precise statement is that any translationally invariant Gibbs
measure is a convex combination of the two Gibbs measures obtained
with plus and minus boundary conditions and that their magnetizations
converge to the magnetizations of the two
pure phases in the van der Waals theory.
In the course of the proof we introduce two ensembles of Gibbs
measures obtained by taking thermodynamic limits with ``mostly" $+$,
respectively $-$, spins in the boundary conditions. We then prove
uniqueness of Gibbs states in each one of these restricted ensembles
by extending the Dobrushin condition for the absence of phase
transitions, [\rcite {dob}], following a coupling scheme introduced
by van den Berg and Maes, [\rcite {BM}], see also
[\rcite {B}], [\rcite {BS}] and [\rcite {GM1}] for other versions
of the coupling and applications.
In the next section we give the main definitions and results, the
proofs are reported in Sections 3 and 4.
\goodbreak
\vskip1truecm
\centerline{\bf 2. Main definitions and results.}
\vskip.5cm
\numsec= 2
\numfor= 1
\centerline {\it Ising systems with Kac potentials.}
We consider an Ising system in $\Bbb Z^d$, $d\ge 2$.
${\Cal X} = \{-1,1\}^{\Bbb Z^d}$ is
the space of all the spin configurations
$\s$ and for any
subset $\L$ of $\Bbb Z^d$, $\s_\L\in
{\Cal X}_\L = \{-1,1\}^\L$ is the restriction
of $\s$ to $\L$.
We denote by
$$
\nu_{\g,\L}(\s_\L|\tau) = Z_{\g,\L}(\tau)^{-1}
\exp\big[-\b H_\g(\s_\L|\tau)\big]
\Eq(2.1)
$$
the Gibbs distribution at the inverse temperature $\b$, in a
finite region $\L$ with boundary conditions
$\tau\in
[-1,1]^{\L^c}$, $\L^c$ the
complement
of $\L$ in $\Bbb Z^d$, (notice that
the outside spins
are allowed to take values in the whole interval $[-1,1]$).
$Z_{\g,\L}(\tau)$ is the partition function (i.e. the
normalization constant which makes $\nu_{\g,\L}(\cdot|\tau)$
a probability on ${\Cal X}_\L$). The hamiltonian is
$$
H_\g(\s_\L|\tau) = - {1\over 2 } \sum_{\scriptstyle x,y \in \L
\atop \scriptstyle x \ne y} J_\g(x,y) \s(x) \s(y) -
\sum_{\scriptstyle x \in \L \atop \scriptstyle y \in \L^c}
J_\g(x,y) \s(x) \tau(y)
\Eq(2.2)
$$
with $\g$ the Kac scaling parameter
(for
simplicity we take $\g=2^{-n}$, $n\in \Bbb N$) and
$$
J_\g(x,y) \doteq \g^d J(\g|x-y|)
\Eq(2.3)
$$
the Kac potential. We suppose $J(s)$ a non-negative,
smooth function supported by $s \in [0,1]$ and
normalized so that
$$
\int_{\Bbb R^d} \! dr \, J(|r|) = 1
\Eq(2.4)
$$
\goodbreak
\vskip.5cm
\centerline {\it The Lebowitz-Penrose theorem.}
Let $F_\g(\b,m)$
be the free energy density (in the thermodynamic limit)
when $\g$ is positive, with $\b$ the inverse temperature and
$m$ the
magnetization density. In [\rcite{LP}] it is
proved that
$$
\lim_{\g \to 0^+}F_\g(\b,m) = \text{CE} \bigg\{ -
{m^2 \over 2} - \b^{-1} i(m) \bigg\} \doteq F_0(\b,m)
\Eq(2.5)
$$
where $\text{CE}\{f\}$ is the convex envelope of
$f$ and $i(m)$ is the Bernoulli entropy:
$$
i(m) = - {1-m \over 2} \log {1-m \over 2} - {1+m \over 2} \log
{1+ m \over 2}
\Eq(2.6)
$$
$F_0(\b,m)$ is a strictly convex
function of
$m\in [-1,1]$ for $\b\le 1$, while for
$\b>1$ it is flat in the interval
$[- m_\b,+m_\b]$, where $m_\b>0$ solves
$$
m_\b = \tanh \{\b m_\b\}
\Eq(2.7)
$$
The mean field inverse critical temperature is thus
$\b^{\rm mf}_c=1$ and for $\b>1$ there are two
and only two pure (homogeneous) phases with magnetizations
$\pm m_\b$.
\vskip.4cm
Let $\b_{\rm c}(\g)$ be the inverse critical
temperature (in the sense
of the unicity of Gibbs states) for the system at $\g>0$.
In [\rcite{CP}] and in [\rcite{BZ}] it is proved
that $\lim_{\g\to 0^+} \b_{\rm c}(\g) =
\b^{\rm mf}_c$.
Let $\nu_{\g}^{\pm}$ be the translationally invariant
Gibbs states obtained in
the thermodynamic limit from finite volume Gibbs states
with boundary condition $\tau$
identically equal to $1$, respectively $-1$ (their existence
follows from ferromagnetic inequalities).
Our main result is:
\goodbreak
\vskip.5truecm
\noindent {\bf 2.1 Theorem.}
\vskip.2truecm
{\sl For any $\b >1$ there is $\g_\b > 0$ so that
for any $\g \le
\g_\b$ any translationally invariant
Gibbs measure is a convex
combination of $\nu_{\g}^+$ and $\nu_{\g}^-$.}
\vskip.5truecm
Since the translationally invariant Gibbs states are identified to
the (homogeneous) phases, [\rcite {Ruelle}], Theorem 2.1 proves that
like in the van der Waals theory
there are only two pure phases when the temperature is
below the mean field critical value
and $\g$ is small enough. It follows from our estimates, but
also from the analysis in Lebowitz and Penrose, [\rcite{LP}],
that the magnetization of the two pure phases
converges to $\pm m_\beta$ as $\g\to 0^+$, thus completing the
proof of the validity of the van der
Waals theory, as discussed in the Introduction.
\goodbreak
\vskip.5cm
\centerline {\it Contours and the Peierls estimates.}
As in the original work of Lebowitz and Penrose our proofs
are based on a coarse graining procedure that, following
[\rcite {CP}], uses the notion of block spins. These are
defined via empirical spin averages with the help
of two spatial scales.
\vskip.5cm
{\it The partitions ${\Cal D}^{(p)}$.}
We denote by $C^{(p)}$, $p\in \Bbb Z$, the cube in $\Bbb Z^d$
of side $2^p\g^{-1}$ and center the origin and
then call ${\Cal D}^{(p)}=\{C^{(p)}_j; \, j \in \Bbb Z^d \}$
the partition of $\Bbb Z^d$ into the translates of the original
cube $C^{(p)}$. Namely $C^{(p)}_j$ is the cube with center
$(2^p\g^{-1}+1)j$ and side $2^p\g^{-1}$.
\vskip.5cm
{\it Block spins.}
Let $k\in \Bbb N$ and given $x\in \Bbb Z^d$ let $C_j^{(-k)}$
be the cube of the partition ${\Cal D}^{(-k)}$
that contains $x$. We then set
$$
\s^{(k)} (x) \doteq {1\over |C^{(-k)}_j|} \sum_{y \in C^{(-k)}_j} \s(y)
\Eq(2.9)
$$
Given $\zeta>0$ and $\g>0$,
we define for any $h \in \Bbb N$ the block spin variables
$\eta_i^{(h)}$, $i \in \Bbb Z^d$, as
$$
\eta_i^{(h)} = \cases {\pm 1 &if $ \big| \s^{(k)}(x) \mp m_\b \big| < \z $
for all $x
\in C^{(h)}_i$ \cr
0 & otherwise \cr }
\Eq(2.8)
$$
\vskip.5cm
By letting $\zeta$ small and $k$ large, a spin configuration
having $\eta_i^{(h)}=1$ becomes close
on $C_i^{(h)}$ to the function constantly
equal to $+m_\b$. This is the closest possible
analogue of a sup norm, given that
the spins have only values $\pm 1$.
After the introduction of the block spins, we
can follow
Pirogov and Sinai (see [\rcite {PS}],
[\rcite {BZ}],
[\rcite {Z}]) and define the notion of (generalized Ising) contours.
\vskip.5cm
{\it Correct blocks.} A block $C_i^{(h)}$
is positive correct if $\eta_i^{(h)}=1$ and $\eta_j^{(h)}=1$
for all
$C^{(h)}_j$ that are $\star$-connected to $C_i^{(h)}$. Similarly
a block $C_i^{(h)}$
is negative correct if $\eta_i^{(h)}=\eta_j^{(h)}=-1$. $C_i^{(h)}$ is
correct if it is either positive or negative correct; it is incorrect
if it is not correct.
\vskip.5cm
{\it Support of a contour and contours.} The support of a contour is a
maximal,
$\star$-connected component of the
incorrect blocks.
A contour is defined by its support and
the specification of the values of the block spins on the support.
By an abuse of notation we sometimes call contour the support of
the contour.
\vskip.5cm
{\it $\pm$ block spin boundary conditions.}
A ${\Cal D}^{(h)}$-measurable region $\L$ has $+$ ($-$) block
spin boundary conditions
if $\tau$ is such that $|\tau^{(k)}(x)-
m_\b|<\zeta$
(respectively $|\tau^{(k)}(x)+ m_\b|<\zeta$) for all $x\in \L^c$
such that dist$(x,\L) \le \g^{-1}$.
In such a case we write $\nu_{\g,\L}(\cdot|\tau)\in G^{\pm}(\L)$.
\vskip.5cm
{\it Contours in finite regions.} Let $\L$ be a bounded, ${\Cal
D}^{(h)}$-measurable region and $\tau$
a $+$ ($-$) block spin boundary condition. We then use the convention
that all the blocks in $\L^c$ are considered
positive (negative) correct and defining those inside $\L$ according to
the previous rules. Contours are defined consequently and their
supports are contained in $\L$. Limited to Section 4,
where $\nu_{\g,\L}(\cdot|\tau)$ is not necessarily in
$G^{\pm}(\L)$, we use on the contrary the convention
that all the
blocks in $\L^c$ are incorrect, those inside are defined
normally.
\vskip.5cm
The typical configurations
of a state in $G^+(\L)$ for small $\g$ is made of a sea of $+$ block
spins with rare islands of $-$ block spins, very much alike
to the
Ising nearest neighbor system at low temperatures: this is a
consequence of
the following
Peierls estimates, proved in [\rcite {CP}].
\goodbreak
\vskip.5truecm
\noindent {\bf 2.2 Theorem.} \ ([\rcite {CP}])
\vskip.2truecm
{\sl There are functions $\zeta^\star(\beta)>0$,
$k^\star(\beta,\zeta)\in \Bbb N$,
$h^\star(\beta,\zeta,k)\in \Bbb N$, $c'(\beta,\zeta,k,h)>0$
and $c(\beta,\zeta,k,h)>0$ so that the following holds. Let
$\g>0$, $\b>1$,
$\zeta\le \zeta^\star(\beta)$, $k\ge k^\star(\beta,\zeta)$,
$h\ge h^\star(\beta,\zeta,k)$, $\L$ a bounded, ${\Cal
D}^{(h)}$-measurable region, $\nu_{\g,\L}(\cdot|\tau)\in
G^{\pm}(\L)$, $n\ge 1$, $C_{i_j}^{(h)}$, $j=1,...,n$, in $\L$,
a n-tuple of distinct, ${\Cal D}^{(h)}$-measurable
cubes in $\L$. Then, calling $c'= c'(\beta,\zeta,k,h)$,
$c= c(\beta,\zeta,k,h)$, $\eta_i= \eta_i^{(h)}$,
$$
\nu_{\g,\L}\Big(\{C_{i_j}\;\text{is incorrect
for all}\;j=1,..,n\}\Big|\tau\Big) \le \left[c'e^{-c\g^{-d}}\right]^n
\Eq (2.10)
$$
and, for any block $C_{i}$ in $\L$,}
$$
\nu_{\g,\L}\Big(\{\eta_{i}=\mp 1\}\Big|\tau\Big) \le c'e^{-c\g^{-d}}
\Eq (2.11)
$$
\vskip.5truecm
In the sequel we will use
the following corollary of Theorem 2.2 whose simple proof
is omitted.
\goodbreak
\vskip.5truecm
\noindent {\bf 2.3 Corollary.}
\vskip.2truecm
{\sl The bounds \equ(2.10) and \equ (2.11) hold with $\b$, $\zeta$
and $k$ as in Theorem 2.2, $h=h_\g$, where
$2^{h_\g}=\g^{-1}$ (recall that $\g$ is in $\{2^{-n}; n \in \Bbb N\}$)
and with $c$ and $c'$ positive parameters that depend only on
$\b$, $\zeta$
and $k$.}
\vskip.5truecm
{\it Notation.} Unless otherwise stated we will
hereafter take $h=h_\g$ and write $\Cal D$
for ${\Cal D}^{(h_\g)}$, $C_i$ for ${C_i}^{(h_\g)}$
and $\eta_i$ for ${\eta_i}^{(h_\g)}$ referring to the cubes $C_i$ as
``the blocks". Sometimes we simply write $C$ for a block in
$\cal D$.
\vskip.5truecm
By Theorem 2.2 and Corollary 2.3 two Gibbs measures in $G^+(\L)$ are
very similar to each other, they have both a lot of $+$ block spins
with very few $-$'s, but they are much more alike than that, as
proved in the next theorem which is the main ingredient in
the proof of Theorem 2.1.
\goodbreak
\vskip.5truecm
\noindent {\bf 2.4 Theorem.}
\vskip.2truecm
{\sl There are $c>0$ and $\g'>0$
so that for any $\g \le \g'$ the following holds.
Let $K$ be a finite set in $\Bbb Z^d$, $\L$
a bounded, ${\Cal D}$-measurable region that contains $K$ and
$\rho$ the distance of $K$ from the boundary of $\L$.
Then, for any $\s_K$ and any}
$\nu_{\g,\L}(\cdot|\tau) \in
G^{\pm}(\L)$,
$$
\big| \nu_{\g,\L}(\s_K|\tau) - \nu_\g^{\pm}(\s_K)
\big| \le |K| e^{-c \g^2\rho}
\Eq(2.12)
$$
\goodbreak
\vskip1truecm
\centerline{\it Outline of the proof of Theorem 2.1.}
\vskip.5truecm
Using Theorem 2.4 we will adapt to the present
case the proof of Gallavotti and Miracle-Sol\'e, [\rcite{GM}],
on the characterization of the translationally invariant
Gibbs states at low temperatures.
Their starting point (in the context of
ferromagnetic, nearest neighbor Ising
systems) was a representation of the spin configurations
as islands at whose
boundaries the spins are either all
$+1$ or all $-1$. At low temperatures these islands are
typically very large, in the sense that,
fixed a bounded set $K$ in $\Bbb Z^d$, the fraction of its
translates that are ``well inside an island" goes to 1
in the thermodynamic limit. When considering a
translationally invariant Gibbs state we can translate
an
observable without changing its expectation,
so that by exploiting the largeness of the islands and the
convergence (in the thermodynamic limit)
of the Gibbs states with all $+$ or all $-$ boundary
conditions,
it is then possible to conclude that the probability of a configuration
$\s_K$ converges to
the probability of $\s_K$ in the $+$ or $-$ Gibbs states,
weighted by the fraction of $+$ or $-$ islands that contain the
translates of
$K$. The proof that the
islands are as large as required by the above argument uses a Peierls
estimate based on the assumption that the temperature is low.
The statement about the convergence of the conditional Gibbs measures
with $+$ or $-$ boundary conditions
follows from ferromagnetic inequalities.
In our case the Peierls
estimates are proved in Corollary 2.3 and they yield
a good lower
bound on the width of the islands. Here $\g$
small plays the role of the low temperature condition in
[\rcite{GM}]. By Theorem 2.2 we can then
replace (in the thermodynamic limit) the expectation
``inside the island" by the expectation in the
$+$ or $-$ states and conclude the proof of Theorem 2.1
(actually our proof will be a little more devious).
While the first part of the argument is mainly a
combination of the techniques developed in [\rcite {CP}] and
in [\rcite{GM}], the proof of Theorem 2.4 is more intriguing and
maybe original. For this reason we have anticipated it
to the next section where we
prove the weaker version stated in Theorem 3.1.
In Section 4 we characterize the typical spin
configurations in terms of $\pm $ islands and using Theorem 3.1 we
will prove first Theorem 2.1 and then Theorem 2.4, as a consequence
of Theorem 3.1 and Theorem 2.1.
\newpage
%\goodbreak
\vskip1truecm
\centerline{\bf 3. Thermodynamic
limit with $+$ or $-$ block spin conditions.}
\vskip.5truecm
\numsec= 3
\numfor= 1
In this section we prove a weaker version of Theorem 2.4.
Let $\L_n$ be an increasing sequence of $\Cal D$-measurable
cubes and
$\nu_{\g,\L_{n}}(\cdot|\tau^{\pm})$ the Gibbs measures
in $\L_n$ with
boundary conditions
$\tau^{\pm}\equiv \pm m_\b$.
By compactness there are $\mu_\g^{\pm}$ and a sequence $\L_{n_k}$
so that $\nu_{\g,\L_{n_k}}(\cdot|\tau^{\pm})$ converges weakly
to $\mu_\g^{\pm}$.
$\mu_\g^{\pm}$
are evidently Gibbs states on $\cal X$
and $\mu^+_\g\ge \mu^-_\g$, stochastically.
\goodbreak
\vskip.5truecm
\noindent {\bf 3.1 Theorem.}
\vskip.2truecm
{\sl In the same context as in Theorem 2.4,}
\goodbreak
$$
\big| \nu_{\g,\L}(\s_K|\tau) - \mu_\g^{\pm}(\s_K)
\big| \le |K| e^{-c \g^2\rho}
\Eq(3.1)
$$
\vskip.5truecm
{\it Remarks.} The factor $\g^2\rho$ will arise as (proportional
to) the minimal number of blocks $C_i$ in a $\star$-connected path
from $K$ to $\L^c$.
We however expect in \equ (3.1)
a dependence on $\g\rho$ rather than $\g^2\rho$,
i.e. lengths measured in units of the interaction range.
Recall
that with $\g$ fixed, $\rho\to +\infty$ as $\L\nearrow\Bbb Z^d$
so that the right hand side of \equ (3.1) vanishes and
$G^{\pm}(\L)$ shrinks, in the weak topology, to
$\mu_\g^{\pm}$.
To have lighter notation we will hereafter restrict to the $+$ case,
the $-$ case following by the spin flip symmetry of the model.
\vskip.5truecm
The remaining of this section is devoted to the proof of Theorem
3.1. Its ultimate origin lies on the stability
of the two phases $\pm m_\b$ in the limit theory. Let
$$
f(m,\b,h) \doteq - {m^2\over 2} - \b^{-1} i(m) - hm
\Eq(3.2)
$$
be the mean field free energy density with external magnetic field
$h$ (that for $h=0$
becomes equal to the expression in
the argument of the convex envelope in \equ (2.5)). For
$h$ small enough (and $\b>1$) $f$ has two local minima,
$m^{\pm}(\b,h)$ which are stable in the sense that
$$
{\partial m^{\pm}(\b,h) \over \partial h} <1
\Eq(3.3)
$$
Thus, if for instance
we want to increase the magnetization $m_\b= m^+(\b,0)$ by
$\delta>0$ we must apply a field $h$ which is strictly larger than
$\delta$. Thus small magnetization fluctuations in the $+$ state at
$h=0$ are not likely to self-sustain, an indication in favour
of the uniqueness of the $+$ state. The difficulty for
the argument
to apply to the proof of Theorem 3.1 is the presence of
large fluctuations that
do necessarily occur somewhere in the infinite space at $\g>0$ and
for which the above stability analysis does not apply.
A state in $G^+(\Lambda)$ (for $\Lambda$ very large) looks
indeed like the
$+$ state $\nu^+_\g$
with a ``large sea" of $+$ block spins by Theorem 2.2 and Corollary
2.4. In this
sea, like for
$\nu^+_\g$, there are rare islands of $-$ block spins, produced
by the large fluctuations mentioned above. The problem is then
to exclude that their
statistics (in $\nu^+_\g$ and $\nu_{\g,\L}$) remains different also
when $\L\nearrow \Bbb Z^d$: locally the states are very similar
but on a larger scale they may in principle have different
behaviors. We will examine separately these two scales, starting
from the former chosen as
the scale of the blocks, i.e. proportional to $ \g^{-2}$. It is
so small that the probability of $\eta_i <1$ is negligible,
so that the above mean
field stability analysis applies, but at the same time it is
much larger than the range $\g^{-1}$ of the interaction
to have a weak dependence on of the boundaries when we look well
inside the block.
\goodbreak
\vskip1cm
\centerline{\it The Dobrushin coupling.}
\vskip.5truecm
Let $x\in \Bbb Z^d$, $\tau$ a configuration on $\Bbb
Z^d\setminus \{x\}$, $\nu_{\g,x}(\cdot|\tau)$ the Gibbs conditional
distribution given $\tau$;
$\tau_{+}$
the configuration on $\Bbb Z^d$ identical to $\tau$ on $\Bbb
Z^d\setminus \{x\}$
completed by putting $\tau_{+}(x)=1$.
We
denote by $A_x$
the set of configurations
$\tau$ such that $|\tau^{(k)}(y)-m_\b|< \zeta$ for all
$y$ such that dist$(x,y)\le \g^{-1}$, see \equ (2.9) for notation.
\goodbreak
\vskip.5truecm
\noindent {\bf 3.2 Lemma.}
\vskip.2truecm
{\sl For any $\zeta>0$ small enough
and $k$ large enough there are
$\g'>0$, $a<1$ and $r_y \ge 0$, $y\ne 0$, with
$$
\sum_{y\ne 0} r_y \le a <1
\Eq (3.4)
$$
such that for any $x\in \Bbb Z^d$, $\tau$ and $\tau'$ in $A_x$ and
$\g \le \g'$}
$$
\Big|E^{\nu_{\g,x}(\cdot|\tau)} \big[\s(x)\big]
- E^{\nu_{\g,x}(\cdot|\tau')} \big[\s(x)\big] \Big|
\le \sum_{y\ne x} r_{y-x}|\tau(y)-\tau'(y)|
\Eq(3.5)
$$
\goodbreak
\vskip.5truecm
\noindent{\it Proof.}
\vskip.2truecm
By an explicit computation
$$
E^{\nu_{\g,x}(\cdot|\tau)} \big[\s(x)\big] =
\tanh\left\{ \beta \sum_{y\ne x} J_\g(x,y)\tau(y)\right\}
\Eq(3.6)
$$
The l.h.s. of \equ (3.5) is then bounded by
$$
{\b \over \cosh^{2}\big(\b m(\tau,\tau')\big)}
\Big|\sum_y J_\g(x,y) \big(\tau(y)-\tau'(y)\big)\Big|
\Eq (3.7)
$$
where $m(\tau,\tau')$ is a number in the interval with endpoints
$$
\sum_{y\ne x} J_\g(x,y) \tau(y) \;\;
\text{ and}\;\;\sum_{y\ne x} J_\g(x,y) \tau'(y)
$$
Recalling that $\tau$ and $\tau'$ are in $A_x$ the first factor
converges to
$$
{\b \over \cosh^{2}(\b m_\b)} =
{\partial m^+(\beta,h) \over \partial h}\Big|_{h=0} < 1
$$
when $\zeta\to 0$, $k\to \infty$ and $\g \to 0^+$. Since
$$
\sum_{y\ne x} J_\g (x,y) \to 1
$$
as $\g \to 0^+$, we obtain \equ (3.5) from \equ (3.7) by suitably
choosing $\zeta$, $k$ and $\g$.
Lemma 3.2 is proved. \qed
\goodbreak
\vskip.5truecm
The inequality \equ (3.5) recalls conditions
that are well known in the
literature: were it holding for all $\tau$ and $\tau'$, it would be
the Liggett contraction condition for spin flip processes, as well as
the
Dobrushin uniqueness condition for
Gibbs fields.
This indicates two
possible strategies, one dynamical, the other one, which is
followed here, in the context of
equilibrium statistical mechanics. The former, which
looks indeed harder, has, on the other
hand,
interest in its own right and is related to
a line of research proposed by Durrett
and Neuhauser, [\rcite {DN}], who investigated properties
of the Glauber+Kawasaki
processes (which have the extra difficulty that
their invariant measures
are not Gibbsian) in terms of the limit behavior
of their mean field limit,
described by a reaction diffusion equation.
Like in our case there are two locally stable phases, but
the proof in
[\rcite {DN}] requires one of them to be metastable. This is used
to prove that when the two phases are simultaneously present in
different regions of space, then the stable one invades the other
at finite speed and, as a consequence, there is a
unique invariant measure for the spin process.
Such a condition fails in our case where the two phases
are completely symmetric, due to the spin
flip symmetry of
the model, and from this it follows that
the interface dynamics is ruled
by a motion by curvature (as proved in the limit $\g\to 0^+$ on a
suitable space-time scaling, [\rcite {DOPT1}], [\rcite {KS}] and references
therein).
This should be sufficient to prove uniqueness in an ensemble
of measures characterized by
the presence of a majority phase, as this will
propagate at
the expenses of the minority one
(according to the motion by curvature)
but the
competition is
much more balanced and the process very slow
(diffusive): an argument like the one
in [\rcite {DN}] does not seem to
apply, even in the simpler context of Gibbsian states.
The Dobrushin approach is based on the
construction of good couplings between Gibbs measures with
different boundary conditions.
\goodbreak
\vskip.5cm
{\it Some basic notation.}
The space where the coupling is defined is
$\O= {\Cal X}\times {\Cal X}$. We call
$\omega = (\s,\s')$ its elements and
$\eta_i$ and $\eta'_i$ the corresponding block spins.
We write
$\O_\Lambda$ and
$\omega_\Lambda$ when we restrict to a region $\Lambda \subset \Bbb
Z^d$. $\s$ and $\s'$ are respectively the first and the second
marginal of $\omega$ and given two probabilities $\nu$ and $\nu'$ on
$\Cal X$ we say that a probability $Q$ on $\O$ is a coupling, or a
joint representation, of $\nu$ and $\nu'$
if its marginal laws are $\nu$ and $\nu'$. Couplings on $\O_\L$
are defined analogously.
\goodbreak
\vskip.5cm
{\it Couplings of single spin distributions.}
Let $\L = \{x\}$, $\nu = \nu_{\g,x}(\cdot|\tau)$,
$\nu' = \nu_{\g,x}(\cdot|\tau')$.
Since $\s(x)$ takes only the values $\pm 1$,
$\nu_{\g,x}(\cdot|\tau)$ and $\nu_{\g,x}(\cdot|\tau')$
are stochastically ordered and therefore there is a
joint representation $q_{\g,x}(\cdot|\tau,\tau')$ on $\O_{\{x\}}$
which is supported by
$\{\s(x)\ge \s'(x)\}$ if $\nu_{\g,x}(\cdot|\tau)\ge\nu_{\g,x}(\cdot|\tau')$
and by $\{\s(x)\le \s'(x)\}$ otherwise. Thus
$$
E^{q_{\g,x}(\cdot|\tau,\tau')} \big[ |\s(x) - \s'(x)| \big] =
\Big| E^{\nu_{\g,x}(\cdot|\tau)} \big[\s(x)\big] -
E^{\nu_{\g,x}(\cdot|\tau')} \big[\s(x)\big] \Big|
\Eq(3.8)
$$
By \equ (3.5), if both $\tau$ and $\tau'$ are in $A_x$, we have
$$
E^{q_{\g,x}(\cdot|\tau,\tau')} \big[ |\s(x) - \s'(x)| \big]
\le \sum_{y\ne x} r_{y-x} |\tau(y) - \tau'(y)|
\Eq(3.9)
$$
\goodbreak
\vskip.5cm
{\it Extension to Gibbs states in finite regions.}
Following Dobrushin, [\rcite{dob}],
we next want to construct couplings on
$\O_\L$ (with $\L$ a bounded, $\Cal D$-measurable region) where
the first moments satisfy again an
inequality like in \equ (3.9). Once done it,
by iteration we will get a good bound on
the moments $|\s(x)-\s'(x)|$, if $x$
is far from the boundaries of $\L$.
The approach requires uniformity on $\tau$
and $\tau'$ in \equ (3.5) with the coefficients $r_y$
that satisfy \equ(3.4).
This fails in our case if we allow either $\tau$
or $\tau'$ to be not in $A_x$, but we can use
the Peierls estimates \equ (2.11) to prove
that this occurs with small
probability. Let $\Delta$ be
either a block or its union with some of
its $\star$-neighbors. We set
$$
e_\g \doteq 2 \max_{\{\Delta\}} \sup_{\nu_{\g,{\Delta}}
(\cdot|\tau)\in G^{+}({\Delta})}\;
\nu_{\g,{\Delta}} \Big(\left\{\text{there is $C_i\subset {\Delta}$
such
that $\eta_i<1$}\right\} \, \Big| \, \tau\Big) \le c'' e^{-c\g^{-d}}
\Eq(3.10)
$$
as it follows from \equ (2.11) with $c''$ equal to $c'$ times the
maximal number of blocks in $\D$.
\goodbreak
\vskip.5truecm
\noindent {\bf 3.3 Lemma.}
\vskip.2truecm
{\sl In the same context of Lemma 3.2 there exists
a joint representation
$Q_{\g,{\Delta}}(\cdot|\tau,\tau')$ of any two measures
$\nu_{\g,{\Delta}}(\cdot|\tau)$ and
$\nu_{\g,{\Delta}}(\cdot|\tau')$ both
in $G^+({\Delta})$, so that,
for any $x\in {\Delta}$,
$$
E^{Q_{\g,{\Delta}}(\cdot|\tau,\tau')} \big[ |\s(x) - \s'(x)|
\big] \le \sum_{y\ne x} r_{y-x} \,
E^{Q_{\g,{\Delta}}(\cdot|\tau,\tau')} \big[|\s (y) -
\s'(y)|
\big]
+ e_\g
\Eq(3.11)
$$
where $\s(y) = \tau(y)$ and
$\s'(y) = \tau'(y)$
for $y\notin {\Delta}$.}
\goodbreak
\vskip.5truecm
\noindent{\it Proof.}
\vskip.2truecm
Setting $\s=\tau$, $\s'=\tau'$ on
${\Delta}^c$ and given $x\in {\Delta}$ we
denote by $\omega_{x^c}=(\si_{x^c},\si'_{x^c})$
a pair of configurations on
$\Bbb Z^d\setminus \{x\}$
and calling $\omega_1(x)=(\si_1(x),\si_1'(x))$ and
$\omega_2(x)=(\si_2(x),\si_2'(x))$ we suppose that
$\omega_1(x)\ne
\omega_2(x)$.
We then define a jump process on $\O_{{\Delta}}$
where the only
transitions are of the type
$$
\Big(\omega_1(x), \omega_{x^c}\Big)
\longrightarrow \Big(\omega_2(x), \omega_{x^c}\Big)
$$
and have intensity $q_{\g,x}(\omega_2(x)|\omega_{x^c})$, see \equ (3.8).
We denote
by $L$ the generator of this process.
We start the process from
the product of $\nu_{\g,{\Delta}}(\cdot|\tau)$ and
$\nu_{\g,{\Delta}}(\cdot|\tau')$.
Then at any time $t\ge 0$ the law of
the process is a joint representation of these two measures.
By compactness
and continuity, the Cesaro time averaged measure converges as
$t\to +\infty$
to a stationary measure
$Q_{\g,{\Delta}}(\cdot|\tau,\tau')$, which
is also a joint representation. Then for any $x\in {\Delta}$
$$
E^{Q_{\g,{\Delta}}(\cdot|\tau,\tau')} \big[ L|\s(x) - \s'(x)| \big]
=0
$$
After expressing the action of the generator we use \equ(3.8)
and Lemma 3.2
in $A_x$, while the probability of
the complement of $A_x$ is bounded in terms of the marginals of
$Q_{\g,{\Delta}}$ which are the Gibbs measures
$\nu_{\g,{\Delta}}(\cdot|\tau)$ and
$\nu_{\g,{\Delta}}(\cdot|\tau')$, thus obtaining \equ (3.11).
Lemma 3.3 is proved. \qed
\goodbreak
\vskip.5truecm {\it The Dobrushin coupling.} We will call
Dobrushin coupling the probability
$Q_{\g,{\Delta}}(\cdot|\tau,\tau')$ of Lemma 3.3 when both
$\nu_{\g,{\Delta}}(\cdot|\tau)$ and
$\nu_{\g,{\Delta}}(\cdot|\tau')$ are in $G^+({\Delta})$,
otherwise we define $Q_{\g,{\Delta}}(\cdot|\tau,\tau')$ as their
product.
\goodbreak
\vskip.5truecm {\it Remarks.}
There obviously exist several
couplings of
$\nu_{\g,{\Delta}}(\cdot|\tau)$ and
$\nu_{\g,{\Delta}}(\cdot|\tau')$,
it is less evident, but nonetheless true, that
when $\tau \ge \tau'$
there are at least two couplings
supported by the set $\{\s_{{\Delta}}
\ge \s_{{\Delta}}'\}$. One is the Dobrushin
coupling derived as above by starting from a
configuration in $\{\s_{{\Delta}}
\ge \s_{{\Delta}}'\}$. It can be seen that this coupling gives
positive probability to any $\omega_{{\Delta}}$ in such a set.
A different coupling will be considered later for larger regions,
its version in the present context would be
supported by configurations
$\omega_{{\Delta}}$ such that if they have ``a circuit of agreement"
then they agree also inside it. This
coupling could also be defined to preserve the order
and from the above support property it would
be different from the
Dobrushin coupling.
Uniqueness is not evident even in the subclass of couplings
defined as the invariant measures of the Markov process of Lemma 3.3,
(observe in fact that its rates are not all strictly positive
and the chain might not be ergodic).
Unfortunately the process is not reversible which means that the
corresponding DLR equations
(i.e. find a measure which has
$q_{\g,x}(\cdot|\tau,\tau')$
as its conditional probabilities)
do not have solutions
(such a property would make the proof of
Theorem 3.1 much simpler). Due to the lack of reversibility even a simple
change of the rates by factors that only depends on $x$ would then
change the invariant measure,
for instance
a different coupling is (in general)
derived starting from the Markov chain at discrete times with
updates $q_{\g,x}(\cdot|\tau,\tau')$, with the sites $x$
ordered
in some arbitrary way. This was the coupling originally
proposed by Dobrushin in [\rcite{dob}].
\vskip.5cm
We conclude the subsection
with a
corollary of Lemma 3.3, Theorem 3.4 below, whose
simple proof is omitted.
{\it The set $R_\g({\Delta},\omega_B)$.} Let
${\Delta}$ be as above and $B^c$ a bounded set containing ${\Cal
D}$. $R_\g({\Delta},\omega_B)$ is
then the set of all
$\omega_{{\Delta}}=(\s_{{\Delta}},\s'_{{\Delta}})$ such that
whenever there is $x\in {\Delta}$ with $\s_B(y)=\s'_B(y)$ for
all $y$ such that dist$(x,y)\, \le \g^{-1}|\log \g|^2$ then
$\s_{{\Delta}}(x)=\s'_{{\Delta}}(x)$.
\goodbreak
\vskip.5truecm
\noindent {\bf 3.4 Theorem.}
\vskip.2truecm
{\sl There is $c>0$
and $\g'>0$ so that for any
$\g\le\g'$
if
both $\nu_{\g,{\Delta}}(\cdot|\tau)$ and $\nu_{\g,{\Delta}}(\cdot|\tau')$
are in $G^+({\Delta})$ then}
$$
Q_{\g,{\Delta}}\Big(R_\g\big({\Delta},(\tau,\tau')\big)
\Big| \tau, \tau' \Big)
\ge 1 - e^{-c|\log\g|^2}
\Eq(3.13)
$$
\goodbreak
\vskip1truecm
\centerline{\it Possible strategies for an extension to larger regions.}
\vskip.5truecm
To extend our analysis to arbitrarily large regions we will use a
coupling which is a variant of that introduced by van den Berg and
Maes, [\rcite {BM}], to prove uniqueness of Gibbs states in models
where the Dobrushin uniqueness condition is not necessarily
satisfied, see also [\rcite {B}], [\rcite {BS}] and [\rcite {GM1}].
The method is quite flexible and it can be adapted to
cases where uniqueness fails and the interaction
has long range, as here. But
let us proceed by a step at a time discussing first the possible
options open after the analysis of the previous subsection,
as we will see the BM coupling comes out as the most natural,
almost mandatory, choice.
For notational simplicity let us
consider the whole space and two Gibbs measures $\nu$ and $\nu'$,
which are limit points of $\{G^+(\L)\}$ as $\L \nearrow \Bbb
Z^d$,
the aim is to prove that $\nu=\nu'$. Calling $\bar C_0=
C_0\cup \pt_0$, where $\pt_0$ is the corridor outside $C_0$ of
width
$\g^{-1}$, we define the partition
${\Cal T} = \{ \bar C_j$; $j\in \Bbb Z^d\}$, obtained by translates of
$\bar C_0$, writing $\bar C_j= C_j\cup \pt_j$ with obvious meaning
of the symbols.
To extend the Dobrushin
coupling to a
coupling between $\nu$ and $\nu'$ we can for instance
distribute the spins
in the corridors
$\partial_j$ by using the product measure. Once the spins
are specified on the corridors, those in the blocks
$C_j$ become conditionally independent and we can thus
apply simultaneously the Dobrushin coupling. As a result we
have
with large probability
agreement inside most of the cubes $C_j$.
However no matter how many and
how large they are
(which is indeed the case if $\g$ is small)
the region of agreement
is only made of
disconnected islands, since the sets
$C_j$ are separated by the corridors
$\partial_j$.
To build
bridges between the islands we apply repeatedly
the same coupling procedure, but using partitions
${{\cal T}}^{(i)}$ obtained by translations
the original one. At each step $i$ we
apply the
coupling procedure relative to the partition
${{\cal T}}^{(i)}$ starting from the coupling realized at the
step $i-1$. After a suitable number of iterations ($2^d$
if the translations are well chosen) we get to
a final coupling $Q^\star$ such that:
\vskip.5cm
{\it A property of $Q^\star$}. With probability 1 there
exist infinitely many, $\star$-connected, ${\cal D}$-measurable
circuits where the spins $\s$ and $\s'$ agree and such that, given
any bounded set $K$, there is a circuit of agreement that strictly
contains $K$ in its interior.
\vskip.5cm
The proof of this statement is
omitted because we have not been able to proceed further
proving
that $\nu=\nu'$, even though it seems we have got not
too far from it. For instance an apparently
``slightly" stronger property would do the
job:
\vskip.5cm
{\it An unproved stronger property of $Q^\star$}.
With probability 1, the circuits of agreement can be chosen in a
family $\{{\Cal C}_i\}$ of increasing circuits.
\vskip.5cm
Then if $\nu$ and $\nu'$ have
trivial $\s$-algebra at infinity, by the martingale
convergence theorem there is a.s. weak convergence of
$\nu_{\g,{\Cal C}_i^0}(\cdot|\s_{{\Cal C}_i})$ to $\nu$ and of
$\nu_{\g,{\Cal C}_i^0}(\cdot|\s'_{{\Cal C}_i})$ to $\nu'$,
where ${\Cal C}_i \in \{{\Cal C}_i\}$ (the increasing family of
circuits) and ${\Cal C}_i^0$ is the interior of
${\Cal C}_i$.
By the validity of ``the stronger property"
there is a.s. a subsequence ${\Cal C}_{i_j}$ in
$\{{\Cal C}_i\}$ where $\s_{{\Cal C}_{i_j}}=\s'_{{\Cal C}_{i_j}}$,
hence $\nu=\nu'$. Unfortunately our analysis only yields the
``property" and not the ``stronger
property", as it is based on the non existence of a percolating path
of disagreement. Presumably $Q^\star$ does not satisfy the ``stronger
property" which is indeed truly stronger than the property we have
proved.
\vskip.5cm
{\it Strong Markov property.}
An alternative idea, which is the one we actually pursue, relies on
the strong Markov property of Gibbs measures. If we look at $\nu$
and $\nu'$ proceeding from the outside and we find a (thick) circuit
$\Cal C$ where the configurations agree, then by the
strong Markov property the configurations will agree in the whole
${\Cal C}^0$.
In the previous setup we could
choose a circuit of agreement $\Cal C$ looking from the outside,
but (in a sense) the coupling $Q^\star$ is constructed looking also
at the inside and the strong Markov property does not hold. Another
way to say it is that
$Q^\star(\omega_{{\Cal C}^0}|\omega_{{\Cal C}})$,
$\omega_{{\Cal C}}=(\s_{{\Cal C}},\s'_{{\Cal C}})$, is not in
general a joint
representation of the the conditional Gibbs measures given
$\s_{{\Cal C}}$ and $\s'_{{\Cal C}}$. If it were, we would then be in business,
because $\s_{{\Cal C}}=\s'_{{\Cal C}}$ as $\Cal C$ is a circuit of
agreement. The marginals of
$Q^\star(\omega_{{\Cal C}^0}|\omega_{{\Cal C}})$ are not the
conditional Gibbs measures
because of the same pathology
(lack of reversibility) observed in
the Remarks after Lemma 3.3. We can however take advantage of this
same
circumstance in the following way. Given a bounded region $\L$ and
calling
$\omega_{\L^c}=(\s_{\L^c},\s'_{\L^c})$ we could from one side define a
Dobrushin coupling $Q_{\g,\L}(\omega_\L|\omega_{\L^c})$, but we may
also proceed differently. Let $\L = \L_1\cup\L_2$,
$\L_1\cap\L_2=\emptyset$,
$Q_{\g,\L}(\omega_{\L_1}|\omega_{\L^c})$
the marginal of $Q_{\g,\L}(\omega_\L|\omega_{\L^c})$ on $\O_{\L_1}$
and $Q_{\g,\L_2}
(\omega_{\L_2}| \omega_{\L_1},\omega_{\L^c})$ the Dobrushin
coupling on $\O_{\L_2}$, given
$\omega_{\L_1}$ and $\omega_{\L^c}$. Then
$$
Q_{\g,\L}(\omega_{\L_1}|\omega_{\L^c})
Q_{\g,\L_2}
(\omega_{\L_2}| \omega_{\L_1},\omega_{\L^c})
$$
is not (in general) equal to the original coupling
$Q_{\g,\L}(\omega_{\L_1},\omega_{\L_2}|\omega_{\L^c})$ (it would be
if the
latter had the Markov property), yet
it is still a coupling which
may even be better than the original one!
We can in fact choose $\L_1$ and $\L_2$
as functions of $\omega_{\L^c}$ (this is all right
because it only requires to look
at the outside)
in such a way that
$\L_1$ screens
$\L_2$ from the ``bad part" of $\L^c$, where there is disagreement.
Then if there is agreement in that part of $\L_1$ that faces $\L_2$,
$\L_2$ only sees agreement at its boundaries and we can
then take the
coupling on $\O_{\L_2}$ which is supported by the diagonal. If
instead the coupling in $\L_1$ is not succesful we can try again,
splitting $\L_2$ in two regions, and so on. This is the strategy
proposed by van den Berg and Maes that we will follow here
using block spins to reduce to nearest neighbor interactions as in
[\rcite {BM}] and distinguishing between bad and good conditioning
(the dangerous $-$ block spins) to
obtain a uniqueness result in the restricted ensemble of states
obtained as limit points of
$\{G^+(\L)\}$.
\goodbreak
\vskip1truecm
\centerline{\it The BM coupling.}
\vskip.5truecm
Let $\L$ be a bounded, ${\cal D}$-measurable
set. We then take $\nu_{\g,\L}(\cdot|\tau)
\in G^+(\L)$, $\nu_{\g,\L_{n_k}}
(\cdot|\tau^+)$, $\L_{n_k} \supset \L$, is an element of the sequence
that defines $\mu_\g^+$.
Let $\tau'$ be a configuration outside $\L$, chosen
with distribution $\nu_{\g,\L_{n_k}}(\cdot|\tau^+)$, $\tau'(x) =
\tau^+(x)=m_\beta$ for $x \in \L_{n_k}$. Given $\tau$ and $\tau'$,
we want to construct a coupling on $\O_\L$ between the two Gibbs
measures $\nu_1(\s_\L) = \nu_{\g,\L}(\s_\L|\tau)$ and $\nu_2(\s_\L)
= \nu_{\g,\L}(\s_\L|\tau')$. The coupling will be the limit law
(attained after a finite number of steps) of a Markov chain.
\vskip.5truecm
{\it The state space.} The state space of the Markov chain is
$\{\xi : \xi = (B,\o_B)\}$ where $B$ is a ${\cal D}$-measurable set
that contains $\L^c$, $\o_B = (\s_B,\s_B') \in \O_B$ and $\s_B(x) =
\tau(x)$, $\s_B'(x) = \tau'(x)$ for $x \in \L^c$.
\vskip.5truecm
{\it Notation.} We fix (arbitrarily)
a one to one map $n(C)$, from $\{C \in {\cal D} : C\subset \L\}$
onto the first $N$ positive integers, where $N$ is the cardinality of
$\{C \in {\cal D} : C\subset \L\}$. Let $\xi = (B,\o_B)$ and $C \subset \L$.
We say that $C$ is of type a) if there is a block $C_i
\subset B$, $\star$-connected to $C$ and such that
$\eta_i + \eta_i'<2$; $C$ is of type b) if it is not of
type a) and if there is $x \in B$ such that dist$(x,C) \le \g^{-1}$
and $\s(x) \ne \s'(x)$. We denote by $F(\xi)$ a set which is the cube
$C$ of type a) with minimal $n(C)$ if this class is not empty,
otherwise it is the cube of type b) with the minimal $n(C)$ unless
also this set is empty, in which case $F=B^c$.
\vskip.5truecm
{\it The transition probability} of the Markov chain, denoted by
$P(\xi,\xi^\star)$, $\xi = (B,\o_B)$, $\xi^\star =
(B^\star,\o^\star_{B^\star})$ is defined as follows. $P(\xi,\xi)=1$ if
$B= \Bbb Z^d$, if $B \ne \Bbb Z^d$ $P(\xi,\xi^\star)$ is supported
by $\xi^\star = (B^\star,\o^\star_{B^\star})$ with $B^\star = B \cup
F(\xi)$ and $\o^\star_{B^\star}(x) = \o_B(x)$ for all $x \in B$. We
thus only need to specify the conditional distribution of
$\o^\star_{F(\xi)}$ that will be denoted by
$\p_\xi(\o^\star_F)$, $F = F(\xi)$. $\p_\xi(\o^\star_F)$ is a
coupling of the marginals on ${\cal X}_F$ of $\nu_{\g,B^c}(\cdot |
\s_B)$ and $\nu_{\g,B^c}(\cdot |\s_B')$. If $F$ is of type a),
calling $\o^\star_F = (\s^\star_F,(\s^\star_F)')$,
$$
\p_\xi(\o^\star_F) = \nu_{\g,B^c}(\s^\star_F|\s_B)
\nu_{\g,B^c}((\s^\star_F)'|\s_B')
\Eq(3.14)
$$
If $F$ is neither of type a) nor of type b) then $F=B^c$ and
$$
\p_\xi(\o^\star_F) = \nu_{\g,B^c}(\s^\star_F|\s_B)
\text{\bf 1}_{\s^\star_F = (\s^\star_F)'}
\Eq(3.15)
$$
Let finally $F$ be of type b). Denote by $\Delta$ the union of $F$ and
of the cubes $C$ in $B^c$ $\star$-connected to $F$. $\pt{\Delta}$ is
the union of all the cubes $C$ not in $\Delta$ but
$\star$-connected to $\Delta$, $\pt^0{\Delta} = \pt{\Delta} \cap
B^c$. Let $p_\xi(\cdot)$ be the probability on $\O_{\pt^0{\Delta}}$
defined as
$$
p_\xi(\tilde \o_{\pt^0{\Delta}}) =
\nu_{\g,B^c}(\tilde\s_{\pt^0{\Delta}}|\s_B)
\nu_{\g,B^c}(\tilde\s_{\pt^0{\Delta}}'|\s_B')
\Eq(3.16)
$$
that is the marginal on $\O_{\pt^0{\Delta}}$ of the product of the
conditional Gibbs measures on ${\Cal X}_{B^c}$. We then set
$$
\pi_\xi(\omega^\star_F)\doteq
E^{p_\xi}\left[ Q_{\g,{\Delta}}(\omega^\star_F
|\tilde\o_{\pt^0{\Delta}},\o_B)\right]
\Eq(3.17)
$$
where $ Q_{\g,{\Delta}}(\cdot
|\tilde\o_{\pt^0{\Delta}},\o_B)$ is the Dobrushin coupling on
$\O_{{\Delta}}$ of the Gibbs measures with boundary conditions
$(\tilde \s_{\pt^0{\Delta}},\s_B)$ and, respectively,
$(\tilde \s'_{\pt^0{\Delta}},\s'_B)$. In \equ (3.17) we then
consider the marginal on $\O_F$ of this coupling (still dependent on
$\tilde\o_{\pt^0{\Delta}}$) which is then integrated by the measure
$p_\xi(\cdot)$ defined in \equ (3.16).
\vskip.5cm
\goodbreak
{\it The van den Berg and Maes (BM) coupling.} Denote by $\Bbb P$
the law and by $\Bbb E$ the expectation of the above Markov chain
$\{\xi_n\}$ when it starts from $\xi_0 = (\L^c,\o_{\L^c})$,
$\o_{\L^c} = (\tau,\tau')$. Let $n\ge N$, $N$ the number of sets
$C$ in $\L$, then
$$
\Bbb P \left( \{\xi_n=\xi_N; \; B_N=\Bbb Z^d\}\right) = 1
\Eq (3.18)
$$
The BM coupling $Q$ is then defined as
$$
Q(\o)=\Bbb P\big(\xi_N=(\Bbb Z^d,\o)\big)
\Eq (3.19)
$$
and since $\o(x)=\big(\tau(x),\tau'(x)\big)$ for all $x\in \L^c$,
$Q(\o)$ may be identified to a probability on $\O_\L$ which
in the next lemma is proved to be a coupling of $\nu_1$ and $\nu_2$.
\goodbreak
\vskip.5cm
\noindent {\bf 3.5 Lemma.}
\vskip.2truecm
{\sl $Q$ is a coupling of $\nu_1$ and $\nu_2$.}
\goodbreak
\vskip.5cm
\noindent{\it Proof.}
\vskip.2truecm
Let $g(\o)$, $\o\in \O$, only depend on $\s_\L$. We need to show
that
$$
E^Q[g] = E^{\nu_{\g,\L}(\cdot|\tau)}
[g]
\Eq (3.20)
$$
the analysis of the other marginal being completely analogous.
Let $\xi=(B,\o_B)$, $\o_B= (\s_B,\s'_B)$ and
$$
G(\xi) \doteq E^{\nu_{\g,B^c}(\cdot|\s_B)}[g]
\Eq (3.21)
$$
We will prove that for all $n\ge 0$
$$
E^Q[g] = \Bbb E[g(\xi_N)] = \Bbb E[G(\xi_n)]
\Eq (3.22)
$$
which proves \equ(3.20): in fact since the Markov chain starts
from
$\xi_0=\big(\L^c,(\tau,\tau')\big)$, $\Bbb E[G(\xi_0)]=
G(\xi_0)$ and by \equ(3.21),
$$
G(\xi_0) = E^{\nu_{\g,\L}(\cdot|\tau)}[g]
$$
which together with \equ (3.22) with $n=0$ proves \equ (3.20).
We
thus only need to prove \equ(3.22). \equ(3.22) is an identity for
$n=N$ because $\xi_N =
(\Bbb Z^d,\o)$ and, by \equ (3.21), $G(\xi_N) = g(\xi_N)$.
It thus remains to prove that
$$
\Bbb E[G(\xi_n)]= \Bbb E[G(\xi_{n-1})]
$$
We have
$$
\Bbb E[G(\xi_n)]= \Bbb E\left[\sum_{\xi^\star}P(\xi_{n-1},\xi^\star)
G(\xi^\star)\right]
\Eq (3.23)
$$
Let $\xi_{n-1}= (B,\o_B)$, then $P(\xi_{n-1},\xi^\star) \ne 0$ only
for $\xi^\star=(B^\star,\o^\star_{B^\star})$, $B^\star = B\cup F$,
$F= F(\xi_{n-1})$, and $\o^\star_{B^\star}(x)=\o_B(x)$ for all $x\in B$.
By \equ (3.21), for such $\xi^\star$,
$$
G(\xi^\star)=E^{\nu_{\g,(B^\star)^c}(\cdot|\s^\star_{B^\star})}[g]
\doteq h(\s^\star_F;\s_B)
\Eq (3.24)
$$
Thus the sum in \equ (3.23) involves only the dependence of $h$ on
its first variable $\s^\star_F$. We have
$$
\sum_{\xi^\star}P(\xi_{n-1},\xi^\star)
G(\xi^\star) =
E^{\nu_{\g,B^c}(\cdot|\s_B)}[h(\s^\star_F;\s_B)]
\Eq (3.25)
$$
because the transition probability
$P(\xi_{n-1},\cdot)$ has been defined as a joint representation on
$\O_F$ of the marginals on ${\Cal X}_F$ of the Gibbs conditional
probabilities on ${\Cal X}_{B^c}$ given $\s_B$ and $\s'_B$ and,
since $\s_B$ is fixed, $h$ is a function on ${\Cal X}_F$. By the
Gibbs property, the r.h.s. of \equ (3.25) becomes
$$
E^{\nu_{\g,B^c}(\cdot|\s_B)}\left[
E^{\nu_{\g,B^c\setminus F}(\cdot|\s^\star_F,\s_B)}[g]\right] =
E^{\nu_{\g,B^c}(\cdot|\s_B)}[g]= G(\xi_{n-1})
$$
hence \equ(3.22). Lemma 3.5 is proved. \qed
\vskip1truecm
\goodbreak
\centerline{{\it A percolation problem.}}
\vskip.5truecm
As in the van der Berg and Maes paper, [\rcite {BM}], we will bound
the difference between $\nu_{\g,\L}(\cdot|\tau)$ and
$\nu_{\g,\L_{n_k}}(\cdot|\tau^+)$ in terms of a percolation
probability.
\vskip.5truecm
{\it Notation.} We write $Q(\cdot)=Q(\cdot|\tau,\tau')$ to make
explicit the dependence on the boundary conditions. Recall that
$\tau$ is fixed while $\tau'$ is the marginal distribution of
$\nu_{\g,\L_{n_k}}(\cdot|\tau^+)$ on ${\Cal X}_{\L^c}$. $\tau'$ is
then with probability 1 identical to $\tau^+$ on $\L_{n_k}^c$.
It is convenient to consider $Q(\cdot|\tau,\tau')$ also as a
probability on $\O$ supported by configurations $\o=(\s,\s')$ such
that
$(\s,\s')=(\tau,\tau')$ on $\O_{\L^c}$.
For any $K\subset \L$ we have
$$
\Big|\nu_{\g,\L}(\s_K|\tau)-\nu_{\g,\L_{n_k}}(\s_K|\tau^+)
\Big| \le E^{\nu_{\g,\L_{n_k}}(\cdot|\tau^+)}\left[Q(
\s_K\ne \s'_K|\tau,\tau')\right] \doteq Q^\star\left
[ \s_K\ne \s'_K\right]
\Eq (3.26)
$$
where $Q^\star$ is the probability on $\O$ defined by
$$
E^{Q^\star}[f] =E^{\nu_{\g,\L_{n_k}}(\cdot|\tau^+)}\left[E^{Q(\cdot
|\tau,\tau')}[f]\right]
\Eq (3.27)
$$
\vskip.5cm
{\it Further properties of the Markov chain.} With $\Bbb P$
probability 1 there is only one trajectory $(\xi_0,\xi_1,...,\xi_N)$
which ends at $\xi_N$. In fact the chain is just a sequence of
specifications added to $\xi_0$ (which is not random) and that
sequentially specify $\xi_1$, $\xi_2$...till $\xi_N$. The
collection of all such specifications are encoded in $\xi_N$ which
thus allows to reconstruct the whole path of the chain.
Since the BM coupling has the same distribution as $\xi_N$ under
$\Bbb P$, by identifying $\xi_N=(\Bbb Z^d,\o)$ with $\o$ we can
define on $\O$ the variables $B_n$ as those corresponding to
$\xi_n=(B_n,\o_{B_n})$ and $F_n=B_n\setminus B_{n-1}$, the notation
being consistent because $\o_{B_n}$ in $\xi_n=(B_n,\o_{B_n})$ is
also the restriction of $\o$ to $B_n$.
\vskip.5cm
{\it Unsuccessful cubes.} $I_0(\o)$ is the set of cubes
$C\in {{\cal D}}$ (called unsuccessful relative to $\o$)
such that the following holds: there is $n\le N$ so that
$F_n=C$ is of type b) relative to
$\xi_{n-1}=(B,\o_{B})$ and $\o_C \notin
R_\g(C,\o_{B})$, which is defined just before Theorem 3.4.
\vskip.5cm
{\it ${\cal D}$ contours.} A block $C$ is ${{\cal D}}_1$ (${{\cal D}}_2$)
positive, negative, correct, incorrect if it is so relative to the
first (second) marginal of $\o$. $I_1(\o)$ ($I_2(\o)$) denotes the
set of the ${{\cal D}}_1$ (${{\cal D}}_2$) incorrect blocks.
A block $C$ is called ${\cal D}$ incorrect if it is either
${{\cal D}}_1$ or ${{\cal D}}_2$ incorrect. ${\cal D}$ contours are the
maximal $\star$-connected components of the set of the
${\cal D}$ incorrect blocks.
\vskip.5cm
{\it The interior of a set, the internal boundary.}
The complement of $\star$-connected bounded
set $A$ has an unbounded, maximal, $\star$-connected component. Its
complement is called the interior of $A$ and it is denoted by
int$(A)$. The strict interior of
$A$ is the part of int$(A)$ which is not in $A$.
The internal boundary $\delta\L$ of $\L$ is the union of all the
cubes $C\subset \L$ that are $\star$-connected to $\L^c$.
\goodbreak
\vskip.5cm
\noindent {\bf 3.6 Proposition.}
\vskip.2truecm
{\sl If $\{\s_K\ne\s'_K\}$ then
with $Q$ probability 1 there is a
$\star$-connected set which is
union of ${\cal D}$ contours and unsuccessful
cubes and whose interior has non void intersection with both
$K$ and
$\delta\L$.}
\goodbreak
\vskip.5cm
\noindent{\it Proof}.
\vskip.2truecm
Without loss of generality we suppose $K$ union of two or more
cubes $C$.
Postponing the proof of
\goodbreak
\vskip.3cm
\noindent {\bf Statement.}
{\sl If
$\s_K\ne\s'_K$ then, with $Q$
probability 1, there are $({\Cal L}_1,...,{\Cal L}_n)$,
$(C_1,...,C_m)$, $m\le n$, and $({\Cal M}_1,...,{\Cal M}_\ell)$ where:
i) \ ${\Cal L}_k$ is a $\star$-connected set of cubes $C_i$
such that $\eta_i+\eta'_i <2$.
ii) \ ${\Cal M}_k$ is a $\star$-connected set of unsuccessful cubes.
iii) \ The union of all the sets ${\Cal L}_k$,
${\Cal M}_k$ and $C_k$ is a $\star$-connected set with non empty
intersection with $K$ and $\delta \L$.}
\vskip.3truecm
\noindent we observe that
for each ${\Cal L}_k$ there is a minimal contour $\G_k$ which
contains ${\Cal L}_k$ in its interior.
Moreover the sets $C_k$ are contained in
the union of the sets $\G_k$. Thus the set in the statement of
the Proposition is the union of the contours $\G_k$ and
${\Cal M}_k$. We are thus left with:
\vskip.3truecm
\noindent {\it Proof of the Statement.}
A bounded ${\cal D}$-measurable, connected set $\Cal C$
is called a path if ${\Cal C}^c$ has only two connected components.
A path is a path of agreement
for $\o$
if $\s(x)=\s'(x)$ for all $x\in {\Cal C}$ which have distance
$\le \g^{-1}$ from the strict interior of $\Cal C$.
With $Q$ probability 1 if $\o$ has a path of agreement $\Cal C$ then
there is agreement also in the whole int$({\Cal C})$.
A cube $C$ is a successful cube for $\o$
if, for some $n$, $C= F(\xi_n)$,
$C$ is of type b) and it is not unsuccessful. If
$\Cal C$ is a path of
successful cubes then it
is a path of agreement.
Given $\o$ let $n$ be such that $B_{n+1}= \Bbb Z^d$ and $F_n$ not a
single cube $C$, so that $F_n$ is a region of agreement. Then
if $\s_K\ne \s'_K$ there is no path made of successful cubes and
cubes in $F_n$ whose interior contains $K$. Therefore there is a
${\cal D}$-measurable, $\star$-connected set
which has non empty intersection with both $K$ and $\delta \L$
made of cubes $C$ which are either unsuccessful or
$\star$-connected to a non positive cube
$C$.
The Statement follows from the above observations.
Proposition 3.6 is proved. \qed
\vskip.5cm
{\it Notation.} Let ${\Cal A}_K$ be the family of all the collections
$A$ of $\star$-connected cubes in $\L_{n_k}$ such that
$$
\text{int}\left(\cup_{C \in A} C \right) \cap K \ne \emptyset
$$
The set in Proposition 3.6 is the union of the cubes in some
$A \in {\Cal A}_K$ with the further property that
$$
\text{int}\left(\cup_{C \in A} C \right) \cap \delta\L \ne \emptyset
$$
For this reason the number $n$ of cubes in $A$ must be
larger than $L_\g$ with $L_\g = c\g^2$dist$(K,\L^c)$, $c>0$ a constant.
\vskip.5cm
Going back to \equ(3.26) and using Proposition 3.6 we then have
$$
\eqalign{
\Big|\nu_{\g,\L}(\s_K|\tau)& -\nu_{\g,\L_{n_k}}(\s_K|\tau^+)
\Big| \cr & \le
\sum_{n\ge L_\g} \sum_{A \in {\Cal A}_K :
|A|=n}\sum_{\{A_0,A_1,A_2\}}
Q^\star\left(\{\o: A_i\subset I_{i}(\o), \, i = 0,1,2 \}\right)
}
\Eq (3.28)
$$
where the last sum is over all the triples $(A_0,A_1,A_2)$ of
disjoint sets whose union is $A$ and such that
$A_0$ and $A_1$ are in $\L$, and
$A_2$ is in $\L_{n_k}$.
By Cauchy-Schwartz
$$
Q^\star\left(\{\o: A_i \subset I_{i}(\o), \, i = 0,1,2 \}\right)^4
\le \prod_{i=0}^2
Q^\star\left(\{\o: A_i \subset I_{i}(\o) \}\right)
\Eq (3.29)
$$
When $i=1$
($i=2$) the event on the right hand side is measurable w.r.t.
$\s$
($\s'$) and since $Q^\star$ is a coupling the
probability becomes the Gibbs probability on $\L$ conditioned
on $\tau$ and respectively the
Gibbs probability on $\L_{n_k}$ conditioned
on $\tau^+$. By Corollary 2.3 we then have
$$
\prod_{i=1}^2
Q^\star\left(\{\o: A_i \subset I_{i}(\o) \}\right) \le \left[c'
e^{-c\g^{-d}}\right]^{|A_1|+|A_2|}
\Eq (3.30)
$$
We are then left with the estimate of
$Q^\star\left(\{\o: A_i \subset I_{0}(\o) \}\right)$. We need a
preliminary lemma:
\goodbreak
\vskip.5cm
\noindent {\bf 3.7 Lemma.}
\vskip.2truecm
{\sl There are $c$ and $c'$ positive so that, for any $\xi=(B,\o_B)$
with $F\equiv F(\xi)$ a cube
of type b),}
$$
\sum_{\xi^\star}P(\xi,\xi^\star)\text{\bf 1}_{\o^\star_F\notin
R_\g(F,\o_B)} \le c' e^{-c|\log\g|^2}
\Eq (3.31)
$$
\goodbreak
\vskip.5cm
\noindent{\it Proof}.
\vskip.2cm
By definition
$$
\sum_{\xi^\star}P(\xi,\xi^\star)\text{\bf 1}_{\o^\star_F\notin
R_\g(F,\o_B)} = \sum_{\o^\star_F} \pi_\xi(\o^\star_F)
\text{\bf 1}_{\o^\star_F\notin
R_\g(F,\o_B)}
\Eq (3.32)
$$
Calling $\Delta$ and $\pt^0{\Delta}$ the sets that enter in the
definition of $P(\xi,\xi^\star)$, we denote by $\Cal G$ the set of
all
$\tilde \o_{\pt^0{\Delta}}$ such that there is a block $C_i$
in $\pt^0{\Delta}$ with $\tilde \eta_i+{\tilde \eta}'_i<2$. Then,
recalling \equ (3.17), the right hand side of \equ (3.32) is
bounded by
$$
p_\xi \left( \tilde \o_{\pt^0{\Delta}} \in {\Cal G} \right) +
\sup_{\tilde \o_{\pt^0{\Delta}} \notin {\Cal G}}
Q_{\g,{\Delta}}\left(\{\o^\star_F\notin
R_\g(F,\o_B)\}
\big|\tilde\o_{\pt^0{\Delta}},\o_B\right)
\Eq (3.33)
$$
${\Cal G} \subset {\Cal G}_1\cup{\Cal G}_2$, where
${\Cal G}_1$ is the set with some $\tilde \eta_i<1$ and
${\Cal G}_2$ the set with some ${\tilde \eta}'_i<1$.
Then, recalling
\equ (3.16) and the Peierls estimate in Corollary 2.3,
the first term in \equ(3.33) is bounded by
$c'e^{-c\g^{-d}}$ for some positive $c'$ and $c$.
The second term in \equ (3.33) is bounded using Theorem 3.4:
in fact,
by the definition of b) cubes, the block spins
on the boundary of $B$ are all positive, hence the assumptions of
Theorem 3.4 are satisfied.
Lemma 3.7 is proved. \qed
\goodbreak
\vskip.5cm
\noindent {\bf 3.8 Proposition.}
\vskip.2truecm
{\sl Let
$c$ and $c'$ be as in Lemma 3.7, then}
$$
Q^\star\Big( A_0 \subset I_0(\o)\Big) \le
\left[c' e^{-c|\log\g|^2}\right]^{|A_0|}
\Eq (3.34)
$$
\goodbreak
\vskip.5cm
\noindent{\it Proof}.
\vskip.2truecm
We fix $\tau'$ in $\L^c$ (the estimates will be independent of
$\tau'$) and
consider the Markov chain $\{\xi_n\}$. Let $\xi_n = (B_n,\o_{B_n})$
and $I_0(\xi_n)$ the set of unsuccessful cubes $C$ that are in
$B_n$. We have
$$
Q\Big( A_0 \subset I_0(\o)\big|\tau,\tau'\Big)
= \Bbb P\Big( A_0 \subset I_0(\xi_N)\Big)
$$
We shorthand $\eps = c'e^{-c|\log\g|^2}$ and, writing
$\xi=(B,\o_B)$, we define
$$
g(\xi) = \eps^{|A_0|- |A_0\cap B|}
\text{\bf 1}_{A_0\cap B\subset I_0(\xi)}
$$
Since $\xi_N= (\Bbb Z^d,\o)$, we have
$$
\Bbb P\Big( A_0 \subset I_0(\xi_N)\Big)= \Bbb E[g(\xi_N)]
$$
We will next prove that for all $n$
$$
\Bbb E[g(\xi_n)] \le \Bbb E[g(\xi_{n-1})]
\Eq(3.35)
$$
In fact
$$
\Bbb E[g(\xi_n)]= \Bbb E\left[\sum_{\xi^\star}P(\xi_{n-1},\xi^\star)
g(\xi^\star)\right]
$$
If $F(\xi_{n-1})\cap A_0 \ne \emptyset$ then
$g(\xi_n)=0$ unless $F(\xi_{n-1})$ is a b) cube $C$ and $\o^\star_C
\notin
R_\g(C,\o_{B_{n-1}})$. Then using Lemma 3.7 the sum over
$\xi^\star$ is bounded by $g(\xi_{n-1})$. If on the other hand
$F(\xi_{n-1})\cap A_0 = \emptyset$, then $g(\xi^\star)=g(\xi_{n-1})$
hence \equ (3.35). Proposition 3.8 is
proved. \qed
\vskip.2truecm
By \equ(3.29), \equ(3.30) and \equ(3.34) we get, for some constant
$\bar c >0$,
$$
Q^\star\left(\{\o: A_i \subset I_{i}(\o), \, i = 0,1,2 \}\right)
\le e^{- \bar c \sum_{i=0}^{2} |A_i| |\log\g|^2}
\Eq(3.36)
$$
Since the number of partitions of a set $A\in {\cal A}_K$
into $(A_1,A_2,A_3)$ is bounded by $3^{|A|}$, from \equ(3.28) and
\equ(3.36) we get, for some $c>0$,
$$
\Big|\nu_{\g,\L}(\s_K|\tau) -\nu_{\g,\L_{n_k}}(\s_K|\tau^+) \Big|
\le \sum_{n \ge L_\g} \left(3 e^{- c |\log\g|^2}\right)^n
N_K(n)
\Eq (3.37)
$$
where $N_K(n)$ is the cardinality of $\{A\in{\cal A}_K : |A|=n\}$.
To estimate $N_K(n)$ we proceed as follows. If $G$ is
any $\star$-connected, ${\cal D}$-measurable region, the number
of its translates whose interior has non void intersection
with $K$ is bounded by $|K|n(G)^d$,
$n(G)^d$ the number of cubes in $G$.
By applying this argument
to the sets in ${\cal A}_K$, we get $N_K(n) \le n^d \tilde N_K(n)$
where $\tilde N_K(n)$ is the number of $\star$-connected,
${\cal D}$-measurable regions $G$ with $n(G) = n$ and which contain a
fixed cube. By using a counting argument in [\rcite{DS}] we get
$\tilde N_K(n) \le a^n$ for some $a>0$. Collecting all the
previous estimates we then get Theorem 3.1.
\vskip1truecm
\goodbreak
\centerline{\bf 4. Proof of Theorem 2.1.}
\vskip.5cm
\numsec= 4
\numfor= 1
In this section we prove Theorem 2.1 following the proof of
Gallavotti and Miracle-Sol\'e for nearest neighbor
interactions, [\rcite {GM}].
We will first prove that Theorem 2.1 follows from Proposition 4.1
below which, together with Theorem 3.1, implies Theorem 2.4.
We will denote in the sequel by $\mu(g)$ the integral of $g$
with respect to $\mu$ while $f$ will always denote
a cylinder function measurable on a bounded set.
\goodbreak
\vskip.5cm
\noindent {\bf 4.1 Proposition.}
\vskip.2truecm
{\sl There is $\g'>0$ so that for any $\g\le\g'$ and
any translationally invariant
Gibbs measure $\nu$ there is $\a
\in [0,1]$ such that
$$
\nu = \a \mu_\g^+ + (1 - \a) \mu_\g^-
\Eq(4.1)
$$
}
\goodbreak
\vskip.5cm
\noindent{\it Proof of Theorems 2.1 and Corollary 2.4}.
\vskip.2truecm
We just need to prove that $\nu^{\pm}_\g= \mu_\g^{\pm}$:
(1)$\;\;
\mu_\g^+\ge \mu_\g^-$, stochastically,
by the definition of $\mu^{\pm}_\g$.
(2)$\;\; \nu^+_\g\ge \mu_\g^+$ since $\nu^+_\g \ge \nu$ for any
Gibbs measure $\nu$; moreover
$\nu^+_\g \le \mu_\g^+$ by \equ(4.1) taking
$\nu= \nu^+_\g$ and using (1). Thus
$\nu^+_\g= \mu_\g^+$. An analogous argument proves that
$\nu^-_\g= \mu_\g^-$.
\qed
\vskip.5cm
\goodbreak
In the sequel
$\L$ will always denote
a ${\Cal D}$-measurable cube with center the origin,
$\tau$ a spin configuration in $\L^c$,
$\nu$ a translationally invariant Gibbs measure,
$\nu_{\L^c}(d\tau)$ the corresponding marginal.
We will first rephrase Proposition 4.1 in a more convenient
way, Proposition 4.2
below, that will then be proved in the
rest of the section. By the DLR equations
$$
\nu(f) = \int \! \nu_{\L^c}(d\tau) \, \nu_{\g,\L}(f|\tau)
\Eq(4.2)
$$
Since
$\nu$ is translationally invariant, for any $a \in \Bbb Z^d$
$$
\nu(f) = \nu(f_a)
\Eq(4.3)
$$
where $f_a$ is $f$ shifted by $a$.
Let ${\Cal A}_\L$ be the set of $a$ for which
$f_a$ is $\s_\L$-measurable
and define
$$
\bar f_\L \doteq {1 \over |{\Cal A}_\L|} \sum_{a \in {\Cal A}_\L} f_a
\Eq(4.4)
$$
Then, since $\nu(f) = \nu(\bar f_\L)$,
$$
\nu(f) = \int \! \nu_{\L^c}(d\tau)
\, \nu_{\g,\L}(\bar f_\L |\tau)
\Eq(4.5)
$$
\goodbreak
\vskip.5truecm
\noindent {\bf 4.2 Proposition.}
\vskip.2truecm
{\sl For any $\g>0$ small enough
there are $\eps_{\g,\L,f}$ and $\a_{\g,\L,\tau}$
such that $\eps_{\g,\L,f}\to 0$ as $\L \nearrow \Bbb Z^d$
for any $\g$ and $f$ fixed; moreover}
$$
\big| \nu_{\g,\L}(\bar f_\L |\tau) - \a_{\g,\L,\tau} \mu_\g^+(f) -
(1 - \a_{\g,\L,\tau})\mu_\g^-(f) \big| \le \eps_{\g,\L,f}
\Eq(4.6)
$$
\vskip.5truecm
\goodbreak
\noindent{\it Proof that Proposition 4.2 implies Proposition 4.1.}
\vskip.2truecm
By \equ (4.6), after integration over $ \nu_{\L^c}(d\tau)$, ($\nu$
a translationally invariant Gibbs measure) we have
$$
\left| \nu(f) -\mu_\g^-(f) - [\mu_\g^+(f) -\mu_\g^-(f)]
\int \! \nu_{\L^c}(d\tau) \a_{\g,\L,\tau} \right| \le \eps_{\g,\L,f}
\Eq(4.7)
$$
By letting $\L\nearrow \Bbb Z^d$ the right hand side vanishes and we
conclude that if $\mu_\g^+=\mu_\g^-$ then $\nu=\mu_\g^-$. But
we know from [\rcite {CP}] that $\nu_\g^{\pm}$ are two
distinct, translationally invariant Gibbs measures, hence $\mu_\g^+
\ne \mu_\g^-$. Thus there is $f$ so that $\mu_\g^+(f)\ne \mu_\g^-(f)$
and we then conclude that
$$
\lim_{\L\nearrow \Bbb Z^d}\int \! \nu_{\L^c}(d\tau) \a_{\g,\L,\tau}
$$
exists. Calling it $\a$ we then obtain \equ(4.1)
as the limit of \equ(4.7) and conclude the proof of Proposition 4.1
which is therefore reduced to that of
Proposition 4.2. \qed
\goodbreak
\vskip.5truecm
The proof of Proposition 4.2 exploits that most of the points $x\in
\L$ are surrounded by a far away connected circuit of blocks $C_i$
where the block spins $\eta_i$ have a definite sign (either all
$+1$ or all
$-1$). By using Theorem 3.1 we then obtain \equ(4.6) with
$\a_{\g,\L,\tau}$ the fraction of points in a plus circuit. The
argument that shows that most of the points are surrounded by the
above circuits is taken from Gallavotti and Miracle-Sol\'e after
generalizing the notion of contour
and using a Peierls estimate proved in [\rcite
{CP}]. Instead of starting from $x$ and looking for circuits that
surround it, we start from the outside and
characterize the region where these circuits do not exist. Such a region
will be an ``external contour", here we
rely on the notion of contours introduced by Pirogov and Sinai
that we adapt to the present context in the following way:
We use the notion of correct blocks with the convention that those
outside $\L$ are incorrect, see the subsection ``Contours and
Peierls estimates" in Section 2.
Each spin configuration $\s_\L$ defines a contour
configuration $\und\G = \{\G_1,...,\G_n\}$. By definition
the set
$$
\pt\L_{\rm int} \doteq \left\{ C_i \subseteq \L \, : \,
\text{dist}(C_i, \L^c) = 0 \right\}
\Eq(4.8)
$$
is incorrect, thus there is an element $\G^{\rm e}$ in
$\und\G$ that contains $\pt\L_{\rm int}$. Let $K^{(1)}$ be the
union of all the blocks $C_i$ in $\L
\setminus \G^{\rm e}$ that are $\star$-connected
to $\G^{\rm e}$ and $K^{(2)}$
the
union of all the blocks $C_i$ in $\L
\setminus [\G^{\rm e}\cup K^{(1)}]$ that are $\star$-connected
to $\G^{\rm e}\cup K^{(1)}$. Let $K_j^{(1)}$,
$j \in J^{(1)}$ and $K_j^{(2)}$,
$j \in J^{(2)}$, be the $\star$-connected components
respectively of $K^{(1)}$ and $K^{(2)}$. We need two circuits
$K^{(1)}$ and $K^{(2)}$ for measurability reasons, see Lemma 4.3
below whose proof is omitted being a straightforward
consequence of the above definitions.
\goodbreak
\vskip.5truecm
\noindent {\bf 4.3 Lemma.}
\vskip.2truecm
{\sl Each one of the sets $K_j^{(2)}$,
$j \in L^{(2)}$, is a closed connected circuit made of blocks
$C_i$ where the block spin $\eta_i$ has a constant,
non zero value. Moreover, calling $\bar\G\doteq\G^{\rm e}\cup K^{(1)}
\cup K^{(2)}$,
$K^{(2)}$ is $\s_{\bar\G}$-measurable.}
\vskip.5truecm
We call $M_j$, $j\in J^{(2)}$, the ``islands" (strictly) inside the
circuits $K_j^{(2)}$. We also denote by $|\G^{\rm e}|$ the number
of cubes $C_i$ that are in $\G^{\rm e}$ and by $L$ the side of
$\L$. Following Gallavotti and Miracle-Sol\'e we distinguish the
cases when $|\G^{\rm e}|0$ so that
in $|\G^{\rm e}|0$ and $f$ fixed.
Going back to \equ(4.9) and using \equ (4.11) we get
$$
\eqalign{
\bigg| \nu_{\g,\L}(&\bar f_\L |\tau) -
E^{\nu_{\g,\L}(\cdot|\tau)}\bigg[
\text{\bf 1}_{|\G^{\rm e}|0$ such that for any
boundary condition $\tau$, any spin configuration $\s_\L$ and any
$\g >0$
$$
\big| H_\g(\s_\L|\tau) - H_\g(\s_\L|m_\b) \big| \le c L^{d-1} \g^{-d}
\Eq(4.14)
$$
Then, recalling the notation $\tau^+\equiv m_\b$, we have
$$
\nu_{\g,\L}
\left( \left\{|\G^{\rm e}|\ge L^{d-2/3}\right\} \Big| \tau \right)
\le e^{2cL^{d-1} \g^{-d}}
\nu_{\g,\L}
\left( \left\{|\G^{\rm e}|\ge L^{d-2/3}\right\} \Big| \tau^+ \right)
\Eq(4.15)
$$
The number of external contours of length $k$ is
bounded by $3^{3kd}$, as it follows from a counting argument
in [\rcite{DS}]. In [\rcite {CP}] it is proved that there is
$c_0>0$ so that
$$
\nu_{\g,\L}\big(\G^{\rm e}\big|\tau^+\big) \le e^{-c_0 \g^{-d}
|\G^{\rm e}|}
\Eq(4.16)
$$
then from \equ(4.15)
$$
\nu_{\g,\L}
\left( \left\{|\G^{\rm e}|\ge L^{d-2/3}\right\} \big| \tau \right)
\le e^{2cL^{d-1} \g^{-d}}\sum_{k=L^{d-2/3}}^\infty
3^{3kd} e^{-c_0 \g^{-d}
k}
$$
which for all $\g$ small enough vanishes as $L\to +\infty$.
We have therefore proved Proposition 4.2. \qed
\goodbreak
\vskip1cm
\centerline{\bf References}
\vskip.5truecm
\item{[\rtag{B}]} J. van den Berg,
{\it A uniqueness condition for Gibbs measures with
application to the two dimensional Ising antiferromagnet}
Commun. Math. Phys. {\bf 152}, 161--166,
(1993)
\vskip.3truecm
\item{[\rtag{BM}]} J. van den Berg, C. Maes,
{\it Disagreement percolation in the study of Markov fields},
Ann. Prob. {\bf 22}, 749--763,
(1994)
\vskip.3truecm
\item{[\rtag{BS}]} J. van den Berg, J.E. Steif,
{\it Percolation and the hard core lattice model},
Stochastic Processes and Appl. {\bf 49}, 179--197,
(1994)
\vskip.3truecm
\item{[\rtag{BP}]} T. Bodineau, E. Presutti,
{\it Phase diagram in Ising systems with additional long range
forces},
CARR Reports in Math. Phys. 3/96,
(1996)
\vskip.3truecm
\item{[\rtag{BZ}]} A. Bovier, M. Zahradnik,
{\it The low temperature phase of Kac-Ising models},
Preprint,
(1996)
\vskip.3truecm
\item{[\rtag{CMP}]} M. Cassandro, R. Marra, E. Presutti,
{\it Upper bounds on the
critical temperature for Kac potentials}, Preprint, (1995)
\vskip.3truecm
\item{[\rtag{CP}]} M. Cassandro, E. Presutti,
{\it Phase transitions in Ising systems with long but finite range},
Markov Processes and Related Fields (to appear)
\vskip.3truecm
\item{[\rtag{DOPT1}]} A. De Masi, E. Orlandi,
E. Presutti, L. Triolo
{\it Glauber evolution with Kac potentials. I.
Mesoscopic and macroscopic limits, interface dynamics},
Nonlinearity {\bf 7}, 1--67, (1994)
\vskip.3truecm
\item{[\rtag{dob}]} R.L. Dobrushin,
{\it Prescribing a system of random variables by conditional
distributions},
Theory Probab. Appl. {\bf 15}, 458--486,
(1970)
\vskip.3truecm
\item{[\rtag{DS}]} R.L. Dobrushin, S. Shlosman,
{\it The problem of translation invariance of Gibbs states at low
temperature},
Soviet Scientific Reviews C, Math. Phys. {\bf 5}, 53--196,
(1985)
\vskip.3truecm
\item{[\rtag{DN}]} R. Durrett, C. Neuhauser,
{\it Particle systems and reaction-diffusion equations},
Annals of Probability, {\bf 22}, 289--333,
(1994)
\vskip.3truecm
\item{[\rtag{GM}]} G. Gallavotti, S. Miracle-Sol\'e,
{\it Equilibrium states of the Ising model in the two-phase region},
Phys. Rev. B, {\bf 5}, 2555--2559,
(1972)
\vskip.3truecm
\item{[\rtag{GM1}]} G. Gielis, C. Maes,
{\it The uniqueness regime of Gibbs fields with unbounded
disorder},
J. Stat. Phys. {\bf 81}, 829--835,
(1995)
\vskip.3truecm
\item{[\rtag{KUH}]} M. Kac, G. Uhlenbeck, P.C. Hemmer,
{\it On the Van der Waals theory of vapour-liquid equilibrium.
I. Discussion of a one dimensional model},
J. Math. Phys. {\bf 4}, 216--228, (1963);
{\it II. Discussion of the distribution functions},
J. Math. Phys. {\bf 4}, 229--247, (1963);
{\it III. Discussion of the critical region},
J. Math. Phys. {\bf 5}, 60--74, (1964)
\vskip.3truecm
\item{[\rtag{KS}]} M.A. Katsoulakis, P.E. Souganidis,
{\it Stochastic Ising models and anisotropic front propagation},
J. Stat. Phys. (to appear)
\vskip.3truecm
\item{[\rtag{LP}]} J. Lebowitz, O. Penrose,
{\it Rigorous treatment of the Van der Waals
Maxwell theory of the liquid vapour transition},
J. Math. Phys. {\bf 7}, 98--113,
(1966)
\vskip.3truecm
\item{[\rtag{PS}]} S.A. Pirogov, Ya. G. Sinai: i) Teor. Mat. Fiz.
{\bf 25}, 358--369, (1975), in Russian; English translation:
{\it Phase diagrams of classical lattice systems},
Theor. Math. Phys. {\bf 25}, 1185--1192, (1975). ii)
Teor. Mat. Fiz. {\bf 26}, 61--76, (1976), in Russian;
English translation: {\it Phase diagrams of classical lattice
systems. Continuation}, Theor. Math. Phys. {\bf 26}, 39--49,
(1976)
\vskip.3truecm
\item{[\rtag{Ruelle}]} D. Ruelle
{\it Statistical mechanics. Rigorous results},
Benjamin, 1969
\vskip.3truecm
\item{[\rtag{Z}]} M. Zahradnik,
{\it A short course on the Pirogov-Sinai theory}
CARR Reports in Math. Phys. 5/96
(1996)
\end