%% !!!!! 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 } } \def\QAND{\quad\hbox{ and }\quad} \def\frac{\over} \def\\{\cr} \def\ref{} \def\nonumber{} \hbox{} \vfill \baselineskip12pt \overfullrule=0in \def\s{\sigma} \def\si{\sigma} \def\eps{\epsilon} \def \ga {\gamma} \def \a {\alpha} \def \ka {\kappa} \def \h {\hat} \def \d {\delta} \def \rh {\rho} \def \ti {\tilde} \def \om {\omega} \def \z {\zeta} \def \I {\Bbb I} \def \ct {\hbox {cost.}} \def \text {\hbox} \def \\\ {\cr} \def \und {\underline} %%%%%%%%References macros start here%%%%%%%%%% % \catcode`\X=12\catcode`\@=11 \def\n@wcount{\alloc@0\count\countdef\insc@unt} \def\n@wwrite{\alloc@7\write\chardef\sixt@@n} \def\n@wread{\alloc@6\read\chardef\sixt@@n} \def\crossrefs#1{\ifx\alltgs#1\let\tr@ce=\alltgs\else\def\tr@ce{#1,}\fi \n@wwrite\cit@tionsout\openout\cit@tionsout=\jobname.cit \write\cit@tionsout{\tr@ce}\expandafter\setfl@gs\tr@ce,} \def\setfl@gs#1,{\def\@{#1}\ifx\@\empty\let\next=\relax \else\let\next=\setfl@gs\expandafter\xdef \csname#1tr@cetrue\endcsname{}\fi\next} \newcount\sectno\sectno=0\newcount\subsectno\subsectno=0\def\r@s@t{\relax} \def\resetall{\global\advance\sectno by 1\subsectno=0 \gdef\firstpart{\number\sectno}\r@s@t} \def\resetsub{\global\advance\subsectno by 1 \gdef\firstpart{\number\sectno.\number\subsectno}\r@s@t} \def\v@idline{\par}\def\firstpart{\number\sectno} \def\l@c@l#1X{\firstpart.#1}\def\gl@b@l#1X{#1}\def\t@d@l#1X{{}} \def\m@ketag#1#2{\expandafter\n@wcount\csname#2tagno\endcsname \csname#2tagno\endcsname=0\let\tail=\alltgs\xdef\alltgs{\tail#2,}% 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\def\t@gsoff#1,{\def\@{#1}\ifx\@\empty\let\next=\relax\else\let\next=\t@gsoff \expandafter\gdef\csname#1cite\endcsname{\relax} \expandafter\gdef\csname#1page\endcsname##1{?} \expandafter\gdef\csname#1tag\endcsname{\relax}\fi\next} \def\verbatimtags{\let\ift@gs=\iffalse\ifx\alltgs\relax\else \expandafter\t@gsoff\alltgs,\fi} \catcode`\X=11 \catcode`\@=\active \localtags %%%%%%%%%%%%references macro end here%%%%%%%%%%%%%%% % %%%%%%%%%%%%%%%%%end of macros%%%%%%%%%%%%%%%%%%%%%% % %%%%%%%%%%%%%%%text starts here%%%%%%%%%%%%%%%%%%%%% \magnification=\magstep1 \hoffset=0.cm \voffset=-.2truecm \hoffset=-.2truecm \hsize=16.5truecm \vsize=24truecm \baselineskip=14pt \parindent=12pt \lineskip=4pt\lineskiplimit=0.1pt\parskip=0.1pt plus1pt \def\ds{\displaystyle}\def\st{\scriptstyle}\def\sst{\scriptscriptstyle} \hyphenation{small} \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. 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