\input amstex \loadbold \define\lnV{\lim_{n\to\infty}\frac 1{|V_n|}} \define\lnn{\lim_{n\to\infty}\frac 1n} \define\lnVl{\lim_{n\to\infty}\frac 1{|V_n|}\log\,} \define\snLa{\sup_{\La\in\RR}\frac 1{|\La|}} \define\nV{\frac 1{|V_n|}} \define\nLa{\frac 1{|\La|}} \define\nVsum{\frac 1{|V_n|}\sum_{\ii\in V_n}} \define\nnsum{\frac 1n\sum_{i=0}^{n-1}} \define\nVl{\frac 1{|V_n|}\,\log\,} \define\lsl{\limsup_{n\to\infty}\,\frac 1{|V_n|}\,\log\,} \define\lil{\liminf_{n\to\infty}\,\frac1{|V_n|}\,\log\,} \define\lsvl#1{\limsup_{n\to\infty}\,\frac1{#1}\,\log\,} \define\livl#1{\liminf_{n\to\infty}\,\frac1{#1}\,\log\,} \define\ldp{large deviation principle} \define\LDP{Large Deviation Principle} \define\lsc{lower semicontinuous} \define\ptrf{Probab. Th. Rel. Fields} \define\cpam{Comm. Pure Appl. Math.} \define\zwvg{Z. Wahrsch. verw. Geb.} \define\superzd{^{\raise1pt\hbox{$\scriptstyle {\Bbb Z}^d$}}} \define\xsubvn{x_{\lower1pt\hbox{$\scriptstyle V(n)$}}} \define\xssubvn{x_{\lower1pt\hbox{$\scriptscriptstyle V(n)$}}} %%%%%%%%%%%% s = how small b = how low \define\ssub#1{_{\lower1pt\hbox{$\scriptstyle #1$}}} \define\ssubb#1{_{\lower2pt\hbox{$\scriptstyle #1$}}} \define\Ssub#1{_{\lower1pt\hbox{$#1$}}} \define\Ssu#1{_{\hbox{$#1$}}} \define\Ssubb#1{_{\lower2pt\hbox{$#1$}}} \define\sssub#1{_{\lower1pt\hbox{$\scriptscriptstyle #1$}}} \define\sssu#1{_{\hbox{$\scriptscriptstyle #1$}}} \define\<{\langle} \define\>{\rangle} \define$${\left(} \define$${\right)} \define${\left[} \define${\right]} \define\lbrak{\bigl\{} \define\rbrak{\bigr\}} \define\lbrakk{\biggl\{} \define\rbrakk{\biggr\}} \define\lbrakkk{\left\{} \define\rbrakkk{\right\}} %%%%%%%%%%%%% s = size p = raise \define\osup#1{^{\raise1pt\hbox{$#1$}}} \define\osupm#1{^{\raise1pt\hbox{$\mskip 1mu #1$}}} \define\osupp#1{^{\raise2pt\hbox{$#1$}}} \define\osuppp#1{^{\raise3pt\hbox{$#1$}}} \define\sosupp#1{^{\raise2pt\hbox{$\ssize #1$}}} \define\sosup#1{^{\raise1pt\hbox{$\scriptstyle \mskip 1mu #1$}}} \define\ssosup#1{^{\raise1pt\hbox{$\scriptscriptstyle #1$}}} \define\ssosupp#1{^{\raise2pt\hbox{$\scriptscriptstyle #1$}}} \define\ssosuppp#1{^{\raise3pt\hbox{$\scriptscriptstyle #1$}}} %%%%measures \define\Ga{\varGamma} \define\La{\varLambda} \define\De{\varDelta} \define\LeD{{\La\erot\De}} \define\jLeD{{\La_j\erot\De_j}} \define\Fi{\varPhi} \define\Fio{\varphi} \define\vPsi{\varPsi} \define\meas#1{\lambda\sosup {#1}} \define\measx{\lambda\sosup {\ox}} \define\measo#1{\lambda_o\sosup {#1}} \define\measox{\lambda_o\sosup {\ox}} \define\measLax{\lambda_\La\sosup {\ox}} \define\Meas#1{\boldsymbol\lambda\sosup {#1}} \define\Measx{\boldsymbol\lambda\sosup {\ox}} \define\Measo#1{\boldsymbol\lambda_o\sosup {#1}} \define\Measox{\boldsymbol\lambda_o\sosup {\ox}} \define\MeasLax{\boldsymbol\lambda_\La\sosup {\ox}} \define\MeasLa#1{\boldsymbol\lambda_\La\sosup {#1}} \define\Mpiessinf{\mme^-_\pi} \define\Mpiesssup{\mme^+_\pi} %%%%% hats , bars \define\Ahat{{\hat A}} \define\Fhat{{\hat F}} \define\Phat{{\hat P}} \define\fhat{{\skew4\hat f}} \define\yhat{{\hat y}} \define\hhat{{\widehat h}} \define\Fbar{{\overline F}} \define\Fubar{{\underline F}} \define\Iubar{{\underline I}} \define\abar{{\overline a}} \define\fbar{{\overline f}} \define\fubar{{\underline f}} \define\jbar{{\overline j}} \define\Kbar{{\overline K}} \define\kbar{{\overline k}} \define\kubar{{\underline k}} \define\Abar{{\overline A}} \define\Xbar{{\overline X}} \define\Bbar{{\overline B}} \define\Cbar{{\overline C}} \define\Ebar{{\overline E}} \define\bbar{{\overline b}} \define\hhbar{{\overline h}} \define\sbar{{\overline s}} \define\Hbar{{\overline H}} \define\Hhat{{\widehat H}} \define\tbar{{\overline t}} \define\Rbar{{\overline R}} \define\rbar{{\overline r}} \define\Pbar{{\overline P}} \define\pbar{{\overline p}} \define\Ubar{{\overline U}} \define\betil{{\widetilde \beta}} \define\nutil{{\tilde \nu}} \define\rhotil{{\tilde \varrho}} \define\thtil{{\tilde\theta}} \define\gamtil{{\tilde\gamma}} \define\gammatil{{\tilde \gamma}} \define\lambdatil{{\tilde \lambda}} \define\sigmatil{{\tilde \sigma}} \define\Jtilde{{\tilde J}} \define\YYtilde{{\widetilde \YY}} \define\OOtilde{{\widetilde \Omega}} \define\xhat{{\hat x}} \define\OObar{{\overline \Omega}} \define\OOhat{{\hat \Omega}} \define\YYhat{{\hat \YY}} \define\Pibar{{\overline \varPi}} \define\psibar{{\overline \psi}} \define\pibar{{\overline \pi}} \define\nubar{{\overline \nu}} \define\gabar{{\overline \gamma}} \define\zebar{{\overline \zeta}} \define\SSbar{{\overline \SS}} \define\KKbar{{\overline \KK}} \define\Ptil{{\widetilde P}} \define\Ctil{{\widetilde C}} \define\Itil{{\widetilde I}} \define\Ftil{{\widetilde F}} \define\Ktil{{\widetilde K}} \define\Htil{{\widetilde H}} \define\Gtil{{\widetilde G}} \define\Rtil{{\widetilde R}} \define\Stil{{\widetilde S}} \define\SStil{{\widetilde \SS}} \define\htil{{\tilde h}} \define\qtil{{\tilde q}} \define\util{{\tilde u}} %%%%% random objects \define\empRn{{\bold R}_n} \define\empRnbar{{\bold {\overline R}}_n} \define\empR{{\bold R}} \define\empRbar{\bold {\overline R}} \define\empRntil{{\bold {\tilde R}_n}} \define\empL{{\bold L}} \define\empLn{{\bold L}_n} \define\empLnbar{{\bold {\overline L}}_n} \define\empMn{{\bold M}_n} %%%%% elements \define\ii{{\bold i}} %\define\aa{{\bold a}} %\define\bb{{\bold b}} \define\ee{{\bold e}} \define\jj{{\bold j}} \define\kk{{\bold k}} \define\bll{{\bold l}} %%%%%% configurations \define\os{{\sigma}} \define\osn{{\os^{(n)}}} \define\ostilde{{\tilde\os}} \define\osLa{\os_\La} \define\osLac{\os_{\La^c}} \define\oo{{\omega}} \define\oz{{\bold z}} \define\om{{\boldsymbol\mu}} \define\onu{{\boldsymbol\nu}} \define\olam{{\boldsymbol\lambda}} \define\oeta{{\boldsymbol\eta}} \define\ox{{\bold x}} \define\oyn{{\oy^{(n)}}} \define\oy{{\bold y}} \define\oytil{{\tilde\oy}} \define\oxtil{{\tilde\ox}} %%%%%%%% controls \define\piste{{\boldsymbol\cdot}} \define\DDD{\displaystyle} \define\iotimes{{\underset \ii\in\bZ^d \to \otimes}\,} \define\iootimes{{\underset \ii \to \otimes}\,} \define\lec{\;\le\;} \define\gec{\;\ge\;} \define\ec{\;=\;} \define\erot{\smallsetminus} \define\symdif{\bigtriangleup} \define\e{\varepsilon} \define\a{\alpha} \define\ett{\,\cdot\,} \define\hhab{h\sosup {a,b}} \define\hhb{h\sosup b} \define\hha{h\sosup a} \define\sbsb{\subset\subset} \define\epi#1{\text{epi}\,#1} \define\kplhyppy{\vskip .2in} \define\pikkuhyppy{\vskip .1in} %%%% sigma fields \define\BXX{{\BB}^\XX} \define\BSS{{\BB}^\SS} \define\BOO{{\BB}^\OO} \define\FSi{\BB^\Sigma} %%%% bold calligraphics \define\FFs{{\bcal F}} \define\LLs{{\bcal L}} \define\LLsbar{{ \overline \LLs}} \define\SSs{{\bcal S}} \define\MMs{{\bcal M}} \define\GGs{{\bcal G}} \define\KKs{{\bcal K}} \define\HHs{{\bcal H}} %%%%% spaces \define\XX{{\Cal X}} \define\OO{\Omega} \define\OX{{\bold X}} \define\OOXX{\boldsymbol\Omega} \define\OOLaXX{\OOXX_\La} \define\XXbar{\overline\XX} \define\OOXXbar{\overline\OOXX} \define\OOLa{\Omega\ssub \La} \define\MOO{\tnmitat {\OO}} \define\MOX{\tnmitat {\OX}} \define\MSOO{\statmitat {\OO}} \define\MSOOSi{\statmitat {\OO\times\Sigma}} \define\MOOSi{\tnmitat {\OO\times\Sigma}} \define\MSSi{\statmitat {\Sigma}} \define\MSSbar{\tnmitat {\SSbar}} \define\mx{\tnmitat {\XX}} \define\ms{\tnmitat {\SS}} \define\msSi{\tnmitat {\SS\times\Sigma}} \define\mz{\tnmitat {\ZZ}} \define\mtx{\statmitat {\XX}} \define\kaikkimitat#1{{\Cal M}(#1)} \define\tnmitat#1{\MM_1(#1)} \define\tnmitatpi#1{\MM_1^\pi(#1)} \define\statmitat#1{\MM\ssub \Theta(#1)} \define\Statmitat#1{\MM\ssub \mmTh(#1)} \define\Statmitatpi#1{\MM\ssub \mmTh^\pi(#1)} %%%%%%% boldfaces \define\mmPhi{\boldsymbol\Phi} \define\mmPsi{\boldsymbol\Psi} \define\mmtheta{{\boldsymbol\theta}} \define\mmTh{{\boldsymbol\Theta}} \define\mmLa{{\boldsymbol\varLambda}} \define\mmS{{\bold S}} \define\mmU{{\bold U}} \define\mms{{\bold s}} \define\mmu{{\bold u}} \define\mmh{{\bold h}} \define\mmF{{\bold F}} \define\mmG{{\bold G}} \define\mmf{{\bold f}} \define\mmw{{\bold w}} \define\mmK{{\bold K}} \define\mmk{{\bold k}} \define\mmg{{\bold g}} \define\mmm{{\bold m}} \define\mme{{\bold e}} %%%%%%%%%%%%%% calligraphics %\define\XX{{\Cal X}} \define\SS{{\Cal S}} \define\cAA{{\Cal A}} \define\BB{{\Cal B}} \define\CC{{\Cal C}} \define\DD{{\Cal D}} \define\EE{{\Cal E}} \define\FF{{\Cal F}} \define\GG{{\Cal G}} \define\HH{{\Cal H}} \define\II{{\Cal I}} \define\LL{{\Cal L}} \define\MM{{\Cal M}} \define\NN{{\Cal N}} \define\PP{{\Cal P}} \define\QQ{{\Cal Q}} \define\RR{{\Cal R}} \define\SSo{{\Cal S}_0} \define\KK{{\Cal K}} \define\UU{{\Cal U}} \define\VV{{\Cal V}} \define\TT{{\Cal T}} \define\WW{{\Cal W}} \define\YY{{\Cal Y}} \define\ZZ{{\Cal Z}} \define\MMtil{{\widetilde \MM}} \define\GGtil{{\widetilde \GG}} %%%%%%%%%%% blackboard bolds, number systems \define\bN{{\Bbb N}} \define\bP{{\Bbb P}} \define\bE{{\Bbb E}} \define\bB{{\Bbb B}} \define\bR{{\Bbb R}} \define\bZ{{\Bbb Z}} \define\bzd{{\Bbb Z}^d} \define\brd{{\Bbb R}^d} \define\brk{{\Bbb R}^k} \define\brplus{{\Bbb R}^+} \define\obN{^{\otimes\bN}} \define\obzd{^{\otimes\bzd}} %%%%%%%%% function spaces \define\cbx{C_b(\XX)} \define\cbs{C_b(\SS)} \define\cbOO{C_b(\OO)} \define\cbOOXX{C_b(\OOXX)} \documentstyle{amsppt} \magnification=\magstep1 %\magnification\magstep2 %\input otsikko \topmatter \title Maximum entropy principles \\ for \\ disordered spins \endtitle \leftheadtext{Timo Sepp\"al\"ainen} \rightheadtext{Maximum entropy principles} \author Timo Sepp\"al\"ainen \endauthor \affil Institute for Mathematics and Its Applications \endaffil \address Institute for Mathematics and Its Applications, Vincent Hall 514, 206 Church St. S.E., University of Minnesota, Minneapolis, MN 55455. Current Address: Institut Mittag-Leffler, Aurav\"agen 17, S-182 62 Djursholm, Sweden \endaddress \email seppalai\@ml.kva.se \endemail \date November 1993 (revised August 1994) \enddate \keywords Large deviations, maximum entropy, quenched disorder, critical exponents \endkeywords \subjclass 60F10, 60K35, 82B44 \endsubjclass \endtopmatter \vskip .4in \document \baselineskip=12pt %\input summary %\NoBlackBoxes %\input part0 \subhead Summary \endsubhead We transform nonstationary independent random fields with exponential \break Radon-Nikodym factors and study the asymptotics of the transformed processes. As applications we deduce conditional limit theorems for such random fields, and we study a Curie-Weiss-type mean-field model of a quenched mixed magnetic crystal. This model has quenched site disorder and frustration but non-random coupling constants. We find a continuous phase transition with critical exponents equal to those of the classical Curie-Weiss theory. \vskip .4in \flushpar{\bf 1 \ \ Introduction} \hbox{} \flushpar This paper is a sequel to the study of large deviations of nonstationary processes, or disordered spins, initiated in \cite{Se1} and \cite{Se2}, though acquaintance with these earlier papers is not necessary for reading the present one. We transform the original process by a family of exponential Radon-Nikodym factors, prove that the collection of transformed processes is sequentially compact in a suitable sense, and characterize the limit points. The limit points are solutions to variational principles, hence the term maximum entropy'. In our case the original processes are independent and the modifying Radon-Nikodym factors are functions of the empirical distribution. Thus this is a way of introducing dependence among the previously independent variables. In Section 2 we remind the reader of the setting in \cite{Se1} and state the general theorems about compactness, limit points, and large deviations of the transformed processes. Proofs appear at the end of the paper in Section 5, and the middle sections 3 and 4 are devoted to two applications. Regarding the two tasks mentioned above, compactness and characterization of limit points, our results on the latter are far more complete than on the former. We use only rather soft large deviation techniques and consequently get convergence only in a very weak sense. It is clear that there is room for improvement here. Our methods do give precise information about the asymptotics of empirical averages, which is useful for studying models from statistical mechanics. Section 3 generalizes the following well-known fact: A fair coin-tossing process, conditioned on consistently producing tails on no fewer than, say, $3/4$ of the tosses, behaves asymptotically like a biased coin with probabilities $(\frac14, \frac34)$. More precisely, if $P=(\frac 12\delta_0+ \frac12\delta_1)\obN$ is the law of an infinite sequence of fair coins and $S_n$ is the number of tails in the first $n$ tosses, then the conditioned laws $$P\bigl(\ett\big|\, {S_n}\geqq \tfrac34n\bigr)$$ converge to $(\frac 14\delta_0+ \frac34\delta_1)\obN$ as $n\to\infty$. This convergence holds not only in the weak topology but also in an information-theoretic sense as proved by Csisz\'ar, who termed this phenomenon asymptotic quasi-independence' \cite{Cs2}. We prove analogous results for nonstationary processes, which means that the coin does not have to be the same for each toss. Section 4 presents an application to the statistical mechanics of quenched disorder. We study a mean-field model of a quenched mixed magnetic crystal where atoms of two different types, A and B, are randomly distributed on the lattice sites and their spins interact in a Curie-Weiss-type mean-field fashion. We characterize the limiting Gibbs measures and study the specific magnetization. The model exhibits a continuous phase transition, with nonzero spontaneous magnetization at low temperatures and with critical exponents equal to those of the classical Curie-Weiss theory. We find no spin glass'' phase where an overlap order parameter would be strictly positive without the accompanying spontaneous magnetization. However, the interaction of the A- and B-atoms does produce interesting effects not visible in a pure A- or B-model. The parameters of the model are chosen so that a pure A-model is a ferromagnetic Curie-Weiss with critical temperature 1, and a pure B-model is an antiferromagnetic Curie-Weiss, hence stays in the paramagnetic phase at arbitrarily low temperatures and has no phase transition. But by strengthening the AB-interaction we can induce ferromagnetic ordering in the quenched model at arbitrarily high temperatures despite the resistance to alignment by the antiferromagnetic BB-coupling. And, surprisingly, the spontaneous magnetization of the B-atoms can be strictly greater than that of the A-atoms. While the proofs of section 4 obviously depend on section 2, the results of section 4 are hopefully accessible to the reader independently of the rest of the paper. To our knowledge, maximum entropy principles have not been studied previously for nonstationary processes. Our development owes much to the elegant treatment of uniformly ergodic Markov chains by Bolthausen and Schmock \cite{BS}, as will become clear to the reader who compares the arguments. For further related results see \cite{B}, \cite{CC}, \cite{CCC}, \cite{HR}, \cite{M}, \cite{Sc}, and their references. Thanks are due to an anonymous referee for several improvements in the final version of the paper. \vskip .4in %\input part1 \vfill\break \flushpar {\bf 2 \ \ The setting and the general results} \hbox{} \flushpar Our setting is that of \cite{Se1} with some changes in notation: At each site $\ii$ of $\bzd$ we have a spin $\os_\ii$ with values in a Polish space $\SS$, and $\OO=\SS\superzd$ denotes the space of spin configurations. $\bzd$ acts on $\OO$ by translations: $(\theta_\ii\os)_\jj=\os_{\ii+\jj}$. Additionally, there is a Polish space $\Sigma$ of {\it quenched variables}, as they are called in the physics literature. (In \cite{Se1} we called them simply {\it parameters.}) Elements of $\Sigma$ are denoted by $\ox$ and $\oy$. We assume that $\bzd$ acts on $\Sigma$ by some homeomorphisms $\theta_\ii$ that satisfy $\theta_\ii\circ\theta_\jj=\theta_{\ii+\jj}$. All infinite-volume limits are taken along the sequence of cubes $V_n=[-n,n]^d\cap\bzd$. We assume given a measurable map $\ox\mapsto p^\ox$ from $\Sigma$ into the space $\ms$ of probability measures on $\SS$ (all measurable structures are Borel and all topologies on measures are generated by bounded continuous functions), and form the product measure $P^\ox= \otimes_{\ii\in\bzd}p^{\theta_\ii\ox}$. The goal is to prove asymptotic results for the spin process $(\os_\ii)$ in terms of the empirical distribution and empirical field, defined by $$\empLn=\nVsum\delta_{\os_\ii}$$ and $$\empRn=\nVsum\delta_{\theta_\ii\os}.$$ The laws of these two measure-valued random variables under the probabilities $P^\ox$ were studied in \cite{Se1}. Here we add some new ingredients: Assume given a measurable function $F:\ms\to [-\infty,c]$, where $c$ is a finite constant. Define probability measures $\gamma_n^\ox$ on $\OO$ by setting $$\gamma_n^\ox(f)=\frac 1{Z_n^\ox} P^\ox\bigl(f\,e^{|V_n|\,F(\empLn)} \bigr) \tag 2.1$$ for bounded Borel functions $f$ on $\OO$, where $Z_n^\ox=P^\ox(e^{|V_n|\,F(\empLn)})$ is the appropriate normalizing factor. In case $P^\ox(F(\empLn)=-\infty)=1$, set $\gamma_n^\ox= P^\ox$. Define the measures $\zeta_n^\ox$ by adding spatial averaging inside $\gamma_n^\ox$: $$\zeta_n^\ox(f)=\gamma_n^\ox(\empRn(f)) =\nVsum \gamma_n^\ox(f\circ \theta_\ii).$$ Our motivation for studying these measures comes from two examples: \roster \item Taking $F=\log \bold1_C$ for a set $C$ amounts to studying the spin process conditioned on the events $\{\empLn\in C\}$. This is interesting when the event $\{\empLn\in C\}$ represents a deviation, that is, its $P^\ox$-probability vanishes as $n\to\infty$. \item In statistical mechanical language, $\gamma_n^\ox$ is a Gibbs measure for the spins $\os_\ii$ located on the sites of $V_n$ whose interaction energy is $-|V_n| F(\empLn(\os))$ and whose noninteracting state is described by the measure $P^\ox$. \endroster The goal of this section is to characterize the limit points of the measures $\gamma_n^\ox$ and $\zeta_n^\ox$ as $n\to\infty$. Sections 3 and 4 present applications corresponding to items (1) and (2) above. Proofs for this section are presented in Section 5. We first develop the terminology and assumptions needed for the results. Throughout, $\pi$ denotes an element of the space $\MSSi$ of invariant probability measures on $\Sigma$. Most of the functions, sets, and measures that appear in our definitions and theorems depend on $\pi$, but to keep the notation simple this dependence is suppressed, as it is fairly easy to detect from the context. We say that $\ox$ is {\it generic} for $\pi$ if both $$\pi=\lim_{n\to\infty}\nVsum\delta_{\theta_\ii\ox} \tag 2.2$$ and $$\pi(\oy:P^\oy\in\ett)=\lim_{n\to\infty}\nVsum \delta\Ssub {P^{\theta_\ii\ox}} \tag 2.3$$ hold in the weak topologies of probability measures on $\Sigma$ and $\MOO$, respectively. The left-hand side in (2.3) is the law of the random measure $\oy\mapsto P^\oy$ under $\pi$. (2.3) follows from (2.2) in case $\ox\mapsto p^\ox$ (and hence $\ox\mapsto P^\ox$) is continuous. In general, $\pi$-a.e. $\ox$ is generic for $\pi$ whenever $\pi$ is ergodic. In \cite{Se1}, conditions (2.2) and (2.3) were termed {\it quasiregularity} and {\it $P$-regularity}, respectively. Given $\pi\in\MSSi$, an entropy for $\nu\in\ms$ is defined by $$K(\nu)=\sup_{f\in b\BSS}\lbrakk \nu(f)-\int_\Sigma\log p^\oy(e^f)\, \pi(d\oy)\rbrakk. \tag 2.4$$ $\BSS$ is the Borel field of $\SS$, and $b\cAA$ denotes the space of bounded $\cAA$-measurable functions for any $\sigma$-field $\cAA$. The function $K$ is related to the familiar {\it relative entropy} or {\it Kullback-Leibler information} in the following way: Define the probability measure $\varphi$ on $\SS\times\Sigma$ by $\varphi(ds,d\oy)=p^\oy(ds)\,\pi(d\oy)$. By Theorem 2.13 of \cite{Se1}, $$K(\nu)=\inf_\rho H(\rho\,|\,\varphi),\tag 2.5$$ where the infimum is over $\rho \in\tnmitat{\SS\times\Sigma}$ with marginals $\nu$ and $\pi$, and the relative entropy $H(\rho\,|\,\varphi)$ is defined by $$H(\rho\,|\,\varphi)=\cases \rho\bigl(\log\frac{d\rho}{d\varphi} \bigr) &\text{if \rho\ll\varphi,}\\ \infty &\text{otherwise.} \endcases$$ The upper and lower semicontinuous regularizations of $F$ are defined by $$F^u(\nu)=\inf_{G: \text{G open}, G\ni \nu} \lbrakk\sup_{\rho:\rho\in G} F(\rho)\rbrakk$$ and $$F_\ell(\nu)=\sup_{G: \text{G open}, G\ni \nu} \lbrakk\inf_{\rho:\rho\in G} F(\rho)\rbrakk.$$ To get results for unbounded measurable functions $F$, we need the following technical assumption. It is automatically satisfied if $F$ is bounded and continuous. \proclaim{Assumption A} $P^\ox(F(\empLn)=-\infty)<1$ for large enough $n$, and $$r_1\equiv\inf_{\nu\in\ms}\{K(\nu)-F^u(\nu)\}= \inf_{\nu\in\ms}\{K(\nu)-F_\ell(\nu)\}<\infty. \tag 2.6$$ \endproclaim An important special case is that where the measure-valued random variables $p^{\theta_\ii\ox}$, $\ii\in\bzd$, are i.i.d. This leads us to formulate the following sharpening of Assumption A: \proclaim{Assumption B} $\Sigma=\XX\superzd$ for a Polish space $\XX$, $\,\bzd$ acts on $\Sigma$ by translations, $p^\ox=p^{\ox_\bold0}$ so that $\ox\mapsto p^\ox$ is $\FSi_0$-measurable, and $\pi=\pi_o\obzd$ for some $\pi_o\in\mx$. Furthermore, assume {\rm (2.6)} and that $P^\ox(F(\empLn)=-\infty) <1$ for large enough $n$, $\pi$-a.s. \endproclaim The notation $\FSi_n$ above denotes the $\sigma$-field of $\Sigma$ generated by the $\XX$-valued coordinate variables $(\ox_\ii:\ii\in V_n)$. Since $H(\ett|\,\varphi)$ is \lsc, strictly convex, and has compact sublevel sets (see \cite{DS} or \cite{V}), (2.5) has a unique minimizer $\rho$ whenever $K(\nu)<\infty$. For such a $\nu$ we denote this unique minimizer by $\psi_\nu$. Let $\psi_\nu^\oy$ denote a conditional distribution of $\psi_\nu$ on $\SS$, given $\oy\in\Sigma$. This associates to $\nu$ a $\pi$-a.s. unique map $\oy\mapsto\psi_\nu^\oy$ from $\Sigma$ into $\ms$ that satisfies $\nu=\int \psi_\nu^\oy\,\pi(d\oy)$ and $$K(\nu)= \int H(\psi_\nu^\oy|\,p^\oy)\,\pi(d\oy). \tag 2.7$$ With this map we define the product measure $\varPsi_\nu^\oy(d\os)= \otimes_{\ii\in\bzd} \psi_\nu^{\theta_\ii\oy}(d\os_\ii)$ on $\OO$ and the invariant measure $\vPsi_\nu(d\os,d\oy)= \vPsi_\nu^\oy(d\os)\,\pi(d\oy)$ on $\OO\times\Sigma$, with marginal $j_\infty(\nu)\in\MSOO$ given by $$j_\infty(\nu)=\int \varPsi_\nu^\oy\,\pi(d\oy). \tag 2.8$$ Under Assumption B, let $\varphi_o(ds,dx) =p^x(ds)\,\pi_o(dx)$ on $\SS\times\XX$. We can replace (2.5) by $$K(\nu)=\inf_\rho H(\rho\,|\,\varphi_o),\tag 2.9$$ where the infimum is now over $\rho \in\tnmitat{\SS\times\XX}$ with marginals $\nu$ and $\pi_o$. (Justification follows in Section 5.) Denote the unique minimizer again by $\psi_\nu$. Now we can take $\psi_\nu^\oy=\psi_\nu^{\oy_\bold0}$ to be $\FSi_0$-measurable and $\vPsi_\nu^\oy(d\os)= \otimes_{\ii\in\bzd}\psi_\nu^{\oy_{\ii}}(d\os_\ii)$. It is obvious that $j_\infty(\nu)$ and $\vPsi_\nu$ are i.i.d. Let $$\KK_1=\{\nu\in\ms: K(\nu)-F^u(\nu)=r_1\}.\tag 2.10$$ Assumption A and the properties of $K$ given in Theorem 2.6 of \cite{Se1} imply that $\KK_1$ is a nonempty compact set. It will turn out that $j_\infty$ is a homeomorphism from $\KK_1$ onto a set $\KK_\infty$ of invariant measures on $\OO$, and $\nu\mapsto\vPsi_\nu^\piste$ is continuous on $\KK_1$ in an appropriate sense. In particular, the integral in (2.12) below makes sense. We are ready to state our main result, Theorem 2.11. For background on the last sentence of the theorem, see \cite{Al} on exchangeable measures and their mixing measures. \proclaim{2.11. Theorem} Let $\ox$ be generic for $\pi$ and assume Assumption A. Then the set $\{\zeta_n^\ox\}_{n=1}^\infty$ is tight. Let $Q$ be any limit point. Then $Q$ is shift-invariant, and there is a probability measure $\La$ on $\KK_1$ such that $$Q=\int_{\KK_1} j_\infty(\nu)\,\La(d\nu), \tag 2.12$$ and $\La$ is a limit point of the laws $\gamma_n^\ox(\empLn\in\ett)$. Under Assumption B $Q$ is exchangeable with mixing measure $\La$. \endproclaim \demo{2.13. Remark} In case $\pi$ is ergodic, subsequences of the two sequences of laws appearing in the theorem converge together. More precisely, for any subsequence $\{n_j\}$ the following are equivalent (still assuming Assumption A and that $\ox$ is generic for $\pi$): \roster \item There exists a limit $\DDD{\varPi=\lim_{j\to\infty} \gamma_{n_j}^\ox(\empR_{n_j}\in\ett)}$ in the weak topology of $\tnmitat{\MOO}$. \item There exists a limit $\DDD{\La=\lim_{j\to\infty} \gamma_{n_j}^\ox(\empL_{n_j}\in\ett)}$ in the weak topology of $\tnmitat{\ms}$. \item There exists a limit $\DDD{Q=\lim_{j\to\infty} \zeta_{n_j}^\ox}$ in the weak topology of ${\MOO}$. \endroster When this happens, $\varPi=\La(j_\infty\in\ett)$, and {\rm (2.12)} is the ergodic decomposition of $Q$. In particular, $\varPi$ and $\La$ are uniquely determined by $Q$. \enddemo More precise information on the convergence of the $\zeta_n^\ox$ seems to depend on the particular model at hand. The Hewitt-Savage 0-1 law implies that, under an i.i.d. $\pi$, any convergence that can be seen on a set of positive $\pi$-probability happens almost everywhere: \proclaim{2.14. Proposition} Assume Assumption B. Fix a subsequence $\{n_j\}$. Then there is a Borel function $g: \Sigma\times\MOO\to\{0,1\}$ such that $g(\ox,Q)=1$ iff $Q$ is a limit point of $\{\zeta_{n_j}^\ox\}_{j=1}^\infty$. For each fixed $Q$, $\ox\mapsto g(\ox,Q)$ is invariant under finite permutations of the coordinates of $\ox$, hence $\pi$-a.s. constant. There is a Borel set $\Sigma_0\subset\Sigma$ such that $\pi(\Sigma_0)=1$ and for any $\ox,\oy\in \Sigma_0$, $\{\zeta_{n_j}^\ox\}_{j=1}^\infty$ and $\{\zeta_{n_j}^\oy\}_{j=1}^\infty$ have the same limit points. \endproclaim Next we discuss the convergence of the measures $\gamma_n^\ox$. It turns out fruitful to consider the maps $\gamma^\piste_n:\oy\mapsto\gamma_n^\oy$ instead of individual measures. Let $\MM_\pi^\piste$ be the space of $\pi$-a.s. defined Borel-measurable maps $\varrho^\piste:\oy\mapsto\varrho^\oy$ from $\Sigma$ into $\MOO$. We put a Polish topology on $\MM_\pi^\piste$ by identifying $\varrho^\piste$ with the measure $\pi\otimes\varrho^\piste\equiv \varrho^\oy(d\os)\, \pi(d\oy)\in\MOOSi$. In other words, this topology is generated by the seminorms $$p_f(\varrho^\piste)=\bigl|\int_\Sigma \varrho^\oy(f^\oy)\, \pi(d\oy)\,\bigr|,$$ where $f$ is a bounded continuous function on $\OO\times\Sigma$ and $f^\oy$ is its $\oy$-section. Write $\pi\varrho^\piste$ for the marginal of $\pi\otimes\varrho^\piste$, $\pi\varrho^\piste= \int \varrho^\oy\,\pi(d\oy)$. Let us say $\varrho^\piste$ is invariant if $\varrho^{\theta_\ii\oy}=\varrho^\oy\circ \theta_{-\ii}$ for all $\ii$, $\pi$-a.s. A particular class of elements of $\MM_\pi^\piste$ are the maps $\vPsi_\nu^\piste:\oy\mapsto\vPsi_\nu^\oy$ associated to $\nu\in\KK_1$. We will show later that the map $(\nu,\oy)\mapsto\vPsi_\nu^\oy$ is jointly measurable on $\KK_1\times\Sigma$, so the integral below is legitimate. In this new notation $j_\infty(\nu)=\pi\vPsi_\nu^\piste$ and $\vPsi_\nu =\pi\otimes\vPsi_\nu^\piste$. \proclaim{2.15. Theorem} Assume Assumption B. Then $\{\gamma_n^\piste\}_{n=1} ^\infty$ is relatively compact. Let $\gamma^\piste$ be any limit point. Then $\gamma^\piste$ is invariant. There is a unique measure $\La$ on $\KK_1$ such that $$\gamma^\piste=\int_{\KK_1} \vPsi_\nu^\piste\,\La(d\nu),\tag 2.16$$ and $\La$ is a limit point of the laws $\pi\gamma^\piste(\empLn\in\ett)$. \endproclaim An analogue of Remark 2.13 holds, but we leave the details to the reader. As one would expect, under suitable circumstances a limit point $Q$ of Theorem 2.11 is an expectation of a limit point $\gamma^\piste$ of Theorem 2.15: \proclaim{2.17. Proposition} Assume Assumption B. Suppose that for some fixed subsequence $\{n_j\}$ and $Q\in\MSOO$, $\zeta_{n_j}^\ox\to Q$ as $j\to\infty$ for all $\ox$ on a set of positive $\pi$-measure. Then $\zeta_{n_j}^\ox\to Q$ for $\pi$-a.e. $\ox$, and there exists a $\gamma^\piste\in \MM_\pi^\piste$ such that $\gamma_{n_j}^\piste\to \gamma^\piste$ as $j\to\infty$. Furthermore, the uniquely determined measures $\La$ of {\rm (2.12)} and {\rm (2.16)} coincide, and $Q=\pi\gamma^\piste$. \endproclaim Underlying these results is a large deviation theory for the laws $\gamma_n^\ox(\empRn\in\ett)$ to which we now turn. We need to extend the entropy defined by (2.4) to $Q\in\MSOO$: Set $$K_n(Q)=\sup_{f\in b\BB^\OO_n}\lbrakk Q(f)-\int_\Sigma\log P^\oy(e^f)\,\pi(d\oy)\rbrakk, \tag 2.18$$ and $$k(Q)=\lnV K_n(Q).\tag 2.19$$ $\BB^\OO_n$ is the $\sigma$-field on $\OO$ generated by the spins in the cube $V_n$. The limit in (2.19) exists by Theorem 2.8 in \cite{Se1}. By Theorems 2.1 and 2.2 of \cite{Se1}, $K$ and $k$ are the rate functions that govern the large deviations of the laws $P^\ox(\empLn\in\ett)$ and $P^\ox(\empRn\in\ett)$, respectively, whenever $\ox$ is generic for $\pi$. Let $Q_0\in\ms$ denote the single-spin marginal of a probability measure $Q\in\MSOO$. Let $$r_\infty=\inf_{Q\in\MSOO}\{k(Q)-F^u(Q_0)\},$$ and for all $Q\in\MOO$, set $$I_\infty(Q)=k(Q)-F^u(Q_0)-r_\infty$$ in case $Q$ is invariant and $I_\infty(Q)=\infty$ otherwise. Let $$\KK_\infty=\{Q\in\MSOO: k(Q)-F^u(Q_0)=r_\infty\} =\{I_\infty=0\}$$ and $$\KK_\infty^\piste =\{\varrho^\piste\in\MM_\pi^\piste : k(\pi\varrho^\piste)-F^u((\pi\varrho^\piste)_0) =r_\infty\}.$$ \proclaim{2.20. Lemma} Suppose $r_1<\infty$. Then $r_1=r_\infty$. $I_\infty$ is a tight rate function, meaning that $I_\infty$ is a \lsc\ function from $\MOO$ into $[0,\infty]$ with compact sublevel sets $\{I_\infty\leqq b\}$ for all real $b$. $\KK_\infty$ and $\KK_\infty^\piste$ are nonempty and compact in their respective topologies. Finally, $j_\infty$ is a homeomorphism from $\KK_1$ onto $\KK_\infty$, whose inverse is the projection $Q\mapsto Q_0$, and $\nu\mapsto\vPsi_\nu^\piste$ is a homeomorphism from $\KK_1$ onto $\KK_\infty^\piste$, with inverse $\varrho^\piste\mapsto (\pi\varrho^\piste)_0$. \endproclaim Let $\Ebar$ and $E^\circ$ denote the weak closure and interior, respectively, of a set $E$ of probability measures. Define $$\Iubar(Q)=k(Q)-F_\ell(Q_0)-r_1$$ for $Q\in\MSOO$ and $\Iubar(Q)=\infty$ for noninvariant $Q$. \proclaim{2.21. Theorem} Assume Assumption A and that $\ox$ is generic for $\pi$. Then for any Borel set $E\subset\MOO$, \aligned -\inf_{Q\in E^\circ}\Iubar(Q)&\le \lil \gamma_n^\ox(\empRn\in E)\\ &\le \lsl \gamma_n^\ox(\empRn\in E)\le -\inf_{Q\in\Ebar}I_\infty(Q). \endaligned \tag 2.22 \endproclaim If $I_\infty=\Iubar$ so that both the upper and the lower bound are given by the same function (this happens whenever $F$ is continuous), we say that a {\ldp} (LDP) holds, or that $I_\infty$ is a rate function for the laws $\gamma_n^\ox(\empRn\in\ett)$. It then follows that the laws $\gamma_n^\ox(\empLn\in\ett)$ satisfy a LDP with rate function $I_1(\nu)=K(\nu)-F^u(\nu)-r_1$, and $\KK_1=\{I_1=0\}$. We close this section with two simple examples to get some feeling for the content of the theorems. They show how measures of the Gibbsian type can arise in our framework. \demo{2.23. Example} The simplest nontrivial example is $F(\nu)=\nu(f)$ for some $f\in\cbs$. Define $\mu_1^\ox,\mu_1\in\ms$ by $d\mu_1^\ox=p^\ox(e^f)^{-1}\,e^f\,dp^\ox$ and $\mu_1=\int\mu_1^\ox\,\pi(d\ox)$. Then $\KK_1=\{\mu_1\}$ and the unique limit point $\gamma^\piste$ of $\gamma_n^\piste$ is defined by $\gamma^\ox=\iootimes\mu_1^{\theta_\ii\ox}$. Indeed, $\gamma_n^\ox$ and $\gamma^\ox$ coincide on $\BOO_n$. \enddemo \demo{2.24. Example} To generalize the previous example, let us study the limit points of the laws $$\gamma^\ox_n(d\os)= \frac 1{Z^\ox_n}\exp\lbrakk |V_n|\,g\biggl( \nVsum f(\os_\ii)\biggr)\rbrakk\,P^\ox(d\os) \tag 2.25$$ for some $f\in\cbs$ and $g\in C_b(\bR)$. To apply Theorem 2.15, set $F(\nu)=g(\nu(f))$. For $\beta\in\bR$ let $d\mu_\beta^\ox=p^\ox(e^{\beta f})^{-1} \,e^{\beta f}\,dp^\ox$ and $\mu_\beta=\int\mu_\beta^\ox\,\pi(d\ox)$. Let $a(\ox)=\text{ess inf }f$ and $b(\ox)= \text{ess sup }f$ with respect to $p^\ox$-measure. Since $\mu_\beta^\ox(f)\to a(\ox)$ as $\beta\to-\infty$, $a$ is measurable, and similarly is $b$. By dominated convergence applied to $\mu_\beta(f)$ as $\beta\to\pm\infty$ it makes sense to adopt the convention $\mu_\infty(f)=\pi(b)$ and $\mu_{-\infty}(f)=\pi(a)$, and this is consistent with setting $$\mu_\infty^\ox=p^\ox(\ett|\,f=b(\ox)) \qquad \text{and}\qquad \mu_{-\infty}^\ox=p^\ox(\ett|\,f=a(\ox)) \tag 2.26$$ whenever these measures are well-defined. Assume $$\pi\{\ox: \text{f is p^\ox-a.s. constant}\} <1.$$ (Otherwise $\KK_1=\{p\}$ where $p=\int p^\ox \,\pi(d\ox)$ and we are done.) Then $$\frac d{d\beta}\mu_\beta(f)=\int \bigl(\mu_\beta^\ox(f^2)- \mu_\beta^\ox(f)^2\bigr)\,\pi(d\ox) >0$$ for real $\beta$, so by dominated convergence $\beta\mapsto\mu_\beta(f)$ is continuous and strictly increasing from $[-\infty,\infty]$ onto $[\pi(a),\pi(b)]$. To compute the entropy $K(\mu_\beta)$, Cor. 3.1 in \cite{Cs1} implies that $\rho=\pi\otimes\mu_\beta^\piste$ is the minimizer for $\nu=\mu_\beta$ in (2.5), hence by the conditional entropy formula ((4.4.8) in \cite{DS} or (10.2) in \cite{V}) $$K(\mu_\beta)=\int H(\mu_\beta^\ox\,|\,p^\ox)\, \pi(d\ox).$$ \demo{Claim} Suppose $\pi(a)\leqq c\leqq \pi(b)$ and assume $\inf_{\nu(f)=c}K(\nu)<\infty$. Then the unique minimizer of $K(\nu)$ subject to $\nu(f)=c$ is $\nu=\mu_\beta$, where $\beta$ is the unique number in $[-\infty,\infty]$ satisfying $\mu_\beta(f)=c$. \enddemo To prove the Claim, suppose first that $c=\pi(a)$. We wish to argue that the existence of a measure $\lambda$ such that $\lambda(f)=c$ and $K(\lambda)<\infty$ implies that the measures $\mu_{-\infty}^\ox$ in (2.26) are well-defined $\pi$-a.s. {}From (2.7), $\psi_\lambda^\ox\ll p^\ox$ and consequently $\psi_\lambda^\ox(f)\geqq a(\ox)$ for $\pi$-a.e. $\ox$. But then $\int \psi_\lambda^\ox(f)\,\pi(d\ox)=c=\pi(a)$ forces $\psi_\lambda^\ox(f=a(\ox))=1$ $\pi$-a.s. It follows that $p^\ox(f=a(\ox))>0$ $\pi$-a.s. so the measures $\mu_{-\infty}^\ox$ are well-defined. On the other hand, we just saw that $\psi_\lambda^\ox(f=a(\ox))=1$ $\pi$-a.s., hence $$H(\mu_{-\infty}^\ox\,|\,p^\ox)\leqq H(\psi_\lambda^\ox|\,p^\ox) \text{\ \ \pi-a.s.}$$ (This is an elementary fact: For any probability measure $\a$ and event $E$ such that $\a(E)>0$, $\rho=\a(\ett|\,E)$ uniquely minimizes $H(\rho\,|\,\a)$ over $\rho$ satisfying $\rho(E)=1$.) Integrating the inequality against $\pi$ gives $K(\mu_{-\infty})\le K(\lambda)$ and proves the Claim for the case $c=\pi(a)$. The case $c=\pi(b)$ is handled similarly. Now let $\pi(a)< c< \pi(b)$ and $\lambda$ again as above. In particular $\beta$ is finite. By the finite-volume variational principle of classical statistical mechanics \cite{Is, p. 46}, $$H(\psi_\lambda^\ox|\,p^\ox)-\beta \psi_\lambda^\ox(f)\geqq H(\mu_\beta^\ox|\,p^\ox)-\beta \mu_\beta^\ox(f)$$ for $\pi$-a.a. $\ox$ with equality iff $\psi_\lambda^\ox= \mu_\beta^\ox$. Integrating against $\pi$ gives $$K(\lambda)-\beta\lambda(f)\geqq K(\mu_\beta) -\beta\mu_\beta(f)$$ with equality iff $\lambda=\mu_\beta$. This proves the Claim. It follows from the Claim that the elements of $\KK_1$ must be $\mu_\beta$'s for some values of $\beta$: For suppose $\lambda\in\KK_1$, that is, $\nu=\lambda$ minimizes $K(\nu)-g(\nu(f))$. Let $c=\lambda(f)$. Then $c=\int \psi_\lambda^\ox(f)\,\pi(d\ox) \in [\pi(a),\pi(b)]$ by the argument used above, and the Claim forces $\lambda=\mu_\beta$ with $\beta$ chosen so that $\mu_\beta(f)=c$. Consequently $\KK_\infty^\piste$ consists of elements $\gamma_\beta^\piste$ defined by $\gamma_\beta^\ox=\iootimes \mu_\beta^{\theta_\ii\ox}$ for each $\mu_\beta$ in $\KK_1$. By Theorem 2.15, the possible limit points of the measures (2.25) are mixtures of the type $$\int_{[-\infty,\infty]} \gamma_\beta^\ox\;\Lambda(d\beta).$$ (Recall though that Theorem 2.15 does not say anything about the limiting behavior of individual measures $\gamma^\ox_n$ but only of the maps $\ox\mapsto\gamma^\ox_n$.) Further conclusions depend on the specific $g$ in question. For example, if $g$ is concave and differentiable on $[\pi(a),\pi(b)]$, then $$\KK_1=\cases \{\mu_\infty\} &\text{if \betag'(\mu_\beta(f)) for all \beta,}\\ \{\mu_{\beta_o}\} &\text{otherwise, where \beta_o=g'(\mu_{\beta_o}(f)) uniquely determines \beta_o.} \endcases$$ \enddemo \vskip .4in %\input part2 \flushpar {\bf 3\ \ Conditional limit theorems} \hbox{} \flushpar In this section we investigate how nonstationary independent random fields behave under conditioning. Such a field is specified by a configuration $\om= (\om_\ii:\ii\in\bzd)$ of probability measures on $\SS$. Put the product measure $P^\om=\otimes_{\ii\in\bzd} \om_\ii$ on $\OO$ so that the spins $(\os_\ii)$ become independent random variables with laws $\LL(\os_\ii)=\om_\ii$. Let $\Sigma=\ms\superzd$ be the space of measure configurations $\om$. Let $\pi\in\MSSi$ with marginal $\pi_o\in\tnmitat\ms$. As in Section 2 let $\varphi_o(ds,d\mu)=\mu(ds)\,\pi_o(d\mu)$, and set $p=\int\mu\, \pi_o(d\mu)$. Throughout this section, $C$ denotes a convex subset of $\ms$ that satisfies $$\inf_{\nu\in\Cbar}K(\nu)=\inf_{\nu\in C^\circ}K(\nu) <\infty.\tag 3.1$$ The first equality in (3.1) is not a vacuous requirement as one can see by constructing simple examples. Assuming that $K(\lambda)<\infty$ for some $\lambda\in C^\circ$, (3.1) holds if \roster \item"{(i)}" $\Cbar=\overline{C^\circ}$ \endroster and \roster \item"{(ii)}" there exists a convex open set $G$ in the ambient space of all signed Borel measures, topologized weakly by $\cbs$, such that $G\cap\ms=C^\circ$. \endroster Under (i) and (ii) $t\mu+(1-t)\lambda \in C^\circ$ for any $\mu\in\Cbar$ and $0\leqq t<1$, and then (3.1) follows from $K$'s convexity. Requirement (i) is trivially true for convex sets with nonempty interior in a locally convex topological vector space, but not so in the relative topology of $\ms$. The convexity requirement of (ii) is also a genuine restriction. Sets that satisfy (i) and (ii) include open and closed $\delta$-fattenings of a given convex set $A$, for $\delta>0$, and intersections of finitely many half-spaces of the form \{\nu\in\ms:\nu(f)-\infty)= P^\om(\empLn\in C)\geqq P^\om(\empLn\in C^\circ),$$which is eventually >0, for by the LDP of Theorem 2.3 in \cite{Se2} and (3.1),$$\lil P^\om(\empLn\in C^\circ)\geqq - \inf_{\nu\in C^\circ}K(\nu) >-\infty.$$Now \KK_1=\{\nu_*\}, so Theorem 2.11 gives Theorem 3.2 and Theorem 2.15 gives Theorem 3.3. \qed \enddemo Let us make some comparisons with the expected process. Suppose \pi is i.i.d. and that \om is generic for \pi (only (2.2) needs to be checked now). The expected process has law P=\int P^\om\,\pi(d\om)= p\obzd. Assume that C satisfies (3.1) not only for K but also for the function H(\ett|\,p) to guarantee that the laws P(\ett|\,\empLn\in C) converge. On the first level of description, the empirical measures \empLn and \empRn behave very much alike under P and P^\om. For example \empLn\to p a.s. under both P^\om and P, and p is the unique zero for both the rate function K of P^\om(\empLn\in\ett) and the rate function H(\ett|\,p) of P(\empLn\in\ett). Only the rate of convergence may differ: K(\nu)\geqq H(\nu\,|\,p) for all \nu (see Theorem 3.9 in \cite{Se2}). However, P(\ett|\,\empLn\in C) is not an average of the laws P^\om(\ett|\,\empLn\in C), and we can no longer expect a common limit. We have P(\ett|\,\empLn\in C)\to\a_*\obzd where \a_* minimizes H(\a\,|\,p) over \a\in \Cbar. There is no a priori reason why \a_* and \nu_* should coincide, and in fact it is easy to construct examples where this does not happen. To clarify the distinction, let us reformulate these minimization problems on the bigger space \SS\times\ms: Let \psi_* minimize H(\rho\,|\,\varphi_o) over probability measures \rho on \SS\times\ms with \rho\ssubb\SS\in \Cbar and \rho\ssubb{\ms}=\pi_o, and let \eta_* minimize H(\rho\,|\,\varphi_o) over \rho with \rho\ssubb\SS\in \Cbar. Then \nu_* and \a_* are the \SS-marginals of \psi_* and \eta_*, respectively, and depending on the particular case, may or may not coincide. In Example 3.5 below \nu_*=\a_* but \psi_*\ne\eta_*, so \psi_*=\eta_* is a stricter requirement. Here is the precise sense: \proclaim{3.4. Proposition} The following are equivalent: \roster \item \psi_*=\eta_*. \item \nu_*=\a_* and K(\nu_*)=H(\a_*\,|\,p). \item\pi_o=\int \varphi_o^s\,\a_*(ds), where \varphi_o^s is a conditional distribution of \varphi_o on \ms, given s\in\SS. \item (\eta_*)_{\ms}=\pi_o. \endroster \endproclaim \demo{Proof} (1)\Longleftrightarrow(4) is immediate since \eta_* minimizes entropy without the constraint (\eta_*)_{\ms}\mathbreak =\pi_o. The minimizing \eta_* must be \eta_*(ds,d\mu)=\a_*(ds)\,\varphi_o^s(d\mu) (look at the conditional entropy formula, (4.4.8) in \cite{DS} or (10.2) in \cite{V}) and consequently H(\a_*|\,p)=H(\eta_*|\,\varphi_o). Hence (3)\Longleftrightarrow(4), and together with K(\nu_*)=H(\psi_*|\,\varphi_o) this gives (1)\Longrightarrow(2). Let \psi_*^s be a conditional distribution of \psi_* on \ms, given s\in\SS. (2) and the conditional entropy formula give$$\aligned K(\nu_*)&=H(\nu_*|\,p)+\int H(\psi_*^s|\,\varphi_o^s) \,\nu_*(ds)\\ &=H(\a_*|\,p)+\int H(\psi_*^s|\,\varphi_o^s) \,\a_*(ds), \endaligned $$hence (2) again forces \int H(\psi_*^s|\,\varphi_o^s) \,\a_*(ds)=0 and we conclude that$$\psi_*=\nu_*(ds)\,\psi_*^s(d\mu) =\a_*(ds)\,\varphi_o^s(d\mu)=\eta_*. \qed$$\enddemo For another point of comparison with the expected process, notice that the measure P(\ett|\,\empLn\in C) not only has identical marginals P(d\os_\ii|\,\empLn\in C) for \ii\in V_n, but also this marginal itself lies in \Cbar. Here is the easy argument: For any f\in b\BB^\SS,$$P(f(\os_\ii)\,|\,\empLn\in C)= P(\empLn(f)\,|\,\empLn\in C)\leqq \sup_{\nu\in C} \nu(f),$$which suffices by the separation theorems of locally convex spaces. The first equality above relies on invariance under permutations which we do not have in our setting. We can make a similar claim only on the average, that is,$$\int P^{\om}(d\os_\ii\,|\,\empLn\in C)\, \pi(d\om)\in \Cbar.$$This follows since$$P^{\tau\om}(f(\os_\ii)\,|\,\empLn\in C)= P^{\om}(f(\os_{\tau\ii})\,|\,\empLn\in C)$$for any permutation \tau on V_n, and \pi is invariant under permutations (still assuming \pi i.i.d.), so$$\int P^{\om}(f(\os_\ii)\,|\,\empLn\in C)\, \pi(d\om)= \int P^{\om}(\empLn(f)\,|\,\empLn\in C)\, \pi(d\om)\leqq \sup_{\nu\in C} \nu(f).$$In particular, it is not clear how a result corresponding to Csisz\'ar's `convergence in information' could be derived in our setting (see p. 790 in \cite{Cs2}). To close this section we return to the coin-tossing example mentioned in the introduction, this time with two different coins. \demo{3.5. Example} Let \frac12t>m(z)\equiv za+(1-z)(1-a), so that the long run frequency m(z) of tails (\equiv the value 1\in\SS) is below t, for almost every realization of the coin choices. Given a typical sequence \om=(\mu_1,\mu_2,\mu_3,\ldots) of coin choices (\mu_i=\a or \beta), how does the coin-tossing process behave upon conditioning on S_n\geqq nt? To be precise, we have \pi_o=z\delta_\a+ (1-z)\delta_\beta, the unique zero of K is p=(1-m(z),m(z)), C=\{\nu:\nu(1)\geqq t\}, and by strict convexity of K, \nu_*=(1-t,t). We need to find \psi^\a and \psi^\beta to produce the \pi_o-a.s. defined map \mu\mapsto \psi^\mu that satisfies \nu_*=\int \psi^\mu\,\pi_o(d\mu) and K(\nu_*)=\int H(\psi^\mu|\,\mu)\,\pi_o(d\mu). We look for it in the form d\psi^\mu=\mu(f)^{-1}\,f\,d\mu, for whenever d\psi=f\otimes g\,d\varphi_o with g(\mu)=\mu(f)^{-1}, \psi must be the minimizer in (2.9) by Cor. 3.1 in \cite{Cs1}. So define a function f on \SS by f(0)=1 and f(1)=\theta for a number \theta to be determined, and set v(\theta)=\int \psi^\mu(1)\,\pi_o(d\mu). Then$$v(\theta)=z\frac{a\theta}{1-a+a\theta}+ (1-z)\frac{(1-a)\theta}{a+(1-a)\theta},$$a strictly increasing continuous function of \theta with v(0)=0 and v(\infty)=1. Let \theta=\theta(t)\in (0,\infty) be the unique number satisfying v(\theta)=t=\nu_*(1); then \psi^\mu is the map we want. This answers our question: The coin-tossing sequence (\mu_1,\mu_2,\mu_3,\ldots), conditioned on S_n\geqq nt, behaves asymptotically like the sequence (\psi^{\mu_1},\psi^{\mu_2}, \psi^{\mu_3},\ldots), where the new coins have probabilities$$\psi^\a=\biggl(\frac{1-a}{1-a+a\theta}\,,\, \frac{a\theta}{1-a+a\theta}\biggr)$$and$$\psi^\beta=\biggl(\frac{a}{a+(1-a)\theta}\,,\, \frac{(1-a)\theta}{a+(1-a)\theta}\biggr).$$Note that \psi^\a and \psi^\beta do depend on z, namely through \theta(t)=\theta(t,z). But for any z it turns out that \a(1)<\psi^\a(1), so perhaps somewhat surprisingly the lowest entropy is achieved by making the \psi^\a-coin even more biased than the \a-coin. (To see this observe that v(1)=m(z) so \theta(t)>1.) Both \partial \psi^\a(1)/\partial z<0 and \partial \psi^\beta(1)/\partial z<0, so picking the \a-coin more and more often decreases the expectations of the new coins. We claimed above (see the paragraph preceding Proposition 3.4) that in this example \a_*=\nu_* but \psi_*\ne \eta_*. That \a_*=\nu_* follows from the one-dimensionality of the example: Both H(\ett|\,p) and K are strictly convex functions with common global minimum at p, so over C they must both be minimized by \nu_*. In particular, the process P^\om(\empRn(\ett)\,|\,S_n\geqq nt) and the conditioned expected process p\obN(\ett|\,S_n\geqq nt) both converge to \nu_*\obN. To see that \psi_*\ne\eta_* take z=\frac12 again and compute$$(\eta_*)_{\ms}=\int \varphi_o^s\,\nu_*(ds) =t(a\delta_\a+(1-a)\delta_\beta)+ (1-t)((1-a)\delta_\a+a\delta_\beta)\ne \pi_o.$$\enddemo \vskip .4in %\input abmodel \flushpar {\bf 4 \ \ A mean-field mixed magnetic crystal} \hbox{} \flushpar Next we apply our theory to a mean-field version of the \text{A}_p\text{B}_{1-p} model of a quenched mixed magnetic crystal. The \text{A}_p\text{B}_{1-p} model was studied by Aharony \cite{Ah} and the mean-field model by Luttinger \cite{Lu}. Our discussion is closer to the analyses of the Curie-Weiss model given by Ellis \cite{E} and Orey \cite{Or}. The quenched model is in fact a generalization of the Curie-Weiss, and turns out to have the same critical behavior. We imagine two types of atoms, A and B, located on the positive integer sites i=1, 2, 3, \ldots and each possessing a \pm 1-valued spin variable \os_i. For each site we flip a coin to place an A-atom with probability p and a B-atom with probability 1-p, where 00 which is another parameter of the model. Let x_i=1 or 0 according to whether site i is occupied by an A- or a B-atom. This gives the quenched variable \ox=(x_i)_{i=1}^\infty. The Hamiltonian or interaction energy of n spins \os=(\os_1,\ldots,\os_n) in the sample \ox with external field h is$$ H^\ox_n(\os)= - \frac 1{2n}\sum_{1\leqq i,j\leqq n}\os_i\os_j \bigl[ x_ix_j -(1-x_i)(1-x_j) +J(x_i(1-x_j)+x_j(1-x_i))\bigr]-h \sum_{1\leqq i\leqq n}\os_i.This Hamiltonian is of the mean-field type in the sense that every spin interacts with every other spin and the strength of the interaction is independent of the distance separating the spins. The model has {\it frustration} in that no spin configuration can simultaneously satisfy all the bonds: In an ABB-triple, the J-coupling tends to align the B spins with the A-spin, but the antiferromagnetic BB-coupling works against aligning the BB-pair. (Frustration requires that x_1,\ldots,x_n not be identical, which happens for large enough n, a.s.) We study the thermodynamic equilibrium of the spins for a fixed realization \ox of the occupation coin flips. This situation is called {\it quenched disorder} in the physics literature. Let \SS=\{\pm 1\} be the single spin space and \XX=\{0,1\}. The space of spin configurations is \OO=\SS^\bN and the space of quenched variables \Sigma=\XX^\bN. The a priori measure of a single spin is \lambda_o=(\delta_{-1}+ \delta_{+1})/2 on \SS, the coin-tossing measure on \XX is \pi_o=p\delta_1+(1-p)\delta_0, and we put the products \lambda=\lambda_o\obN and \pi=\pi_o\obN on \OO and \Sigma, respectively. Once an inverse temperature \beta>0 is specified, the Gibbs measure \mu_n^{\ox} gives probability\mu_n^{\ox} (\os)=\frac1{Z_n^\ox} e^{-\beta\,H_n^\ox(\os)}\lambda_n(\os)\tag 4.1$$to the configuration \os=(\os_1,\ldots,\os_n). Here \lambda_n is the restriction of \lambda to n spins. We shall first describe the results for this model and then present the proofs at the end. The limiting Gibbs measures are determined by the magnetizations of A- and B-atoms, or mathematically speaking, by the expected values of the spins. Let z_1 denote the expected spin of an A-atom and z_0 of a B-atom. The equilibrium values (z_0,z_1)=(z_0(\beta,h), z_1(\beta,h)) that minimize free energy are among the solutions of$$\aligned s_1&= \xi_1(s_0)\equiv \frac 1{\beta Jp} \left[ \frac12 \log\frac{1+s_0}{1-s_0} +\beta(1-p)s_0-\beta h\right] \\ s_0&=\xi_0(s_1)\equiv \frac 1{\beta J(1-p)} \left[ \frac12 \log\frac{1+s_1}{1-s_1} -\beta p s_1-\beta h\right] \endaligned \tag 4.2 $$for (s_0,s_1) in [-1,1]\times[-1,1]. These equations reveal a phase transition at the critical temperature$$\beta_c=\beta_c(p,J)= \frac {1-2p+(1+4J^2p(1-p))^{1/2}} {2(1+J^2)p(1-p)}. \tag 4.3 $$These are the equilibrium solutions of (4.2) for \beta>0 and real h: \roster \widestnumber\item{\beta\leqq\beta_c, h=0:} \item"{\beta\leqq\beta_c, h=0:}" The unique solution (z_0,z_1)=(0,0). \item"{\beta>\beta_c, h=0:}" \{(-z_0,-z_1), (z_0,z_1)\} , where (z_0, z_1) is the unique solution in (0,1)\times(0,1). The solution (0,0) does not qualify as an equilibrium. \item"{\beta>0, \;h>0:}" The unique solution (z_0, z_1) in (0,1)\times(0,1). Other solutions do not qualify as equilibria. \item"{\beta>0, \;h<0:}" The unique solution (z_0, z_1) in (-1,0)\times(-1,0). Other solutions do not qualify as equilibria. \endroster It is clear from (4.2) that (z_0(\beta,-h),z_1(\beta,-h))=(-z_0(\beta,h), -z_1(\beta,h)) for h\ne 0. Next we define measures \mu^\ox on \OO determined by the spin expectations, for a given occupation variable \ox=(x_i)_{i=1}^\infty: \roster \widestnumber\item{\beta\leqq\beta_c and h=0:} \item"{\beta\leqq\beta_c or h\ne 0:}" ({\it The uniqueness case}) Define \psi^x\in\ms by$$\psi^x= \frac{1-z_{x}}2 \delta_{-1}+ \frac{1+z_{x}}2\delta_{+1}\tag 4.4$$for x=0,1 and then \mu^\ox=\otimes_{i=1}^\infty \psi^{x_i}. Note that \mu^\ox=\lambda for all \ox in case \beta\leqq\beta_c and h=0. \item"{\beta>\beta_c and h=0:}" ({\it The nonuniqueness case}) Set \mu^\ox_{\pm}= \otimes_{i=1}^\infty \psi^{x_i}_\pm, where \psi^x_\pm\in\ms are defined by$$\psi^x_\pm= \frac{1\mp z_{x}}2 \delta_{-1}+ \frac{1\pm z_{x}}2\delta_{+1} \tag 4.5for x=0,1. Then put \mu^\ox =(\mu^\ox_++\mu^\ox_-)/2. \endroster This defines an element \mu^\piste =\mu^{\beta,h,\piste} of \MM^\piste_\pi. The connection to the z_x is clear: For example under \mu^\ox_+ the expected spin at site i is z_1 if the site is occupied by an A-atom and z_0 if the site is occupied by a B-atom. Recall the Polish topology on \MM_\pi^\piste defined in Section 2 after Proposition 2.14. \proclaim{4.6. Theorem} For all p, J, \beta, and h, \mu^\piste=\lim \mu^\piste_n as n\to\infty. \endproclaim Considering \mu^\ox as the equilibrium state of the spins with occupations \ox, we can say the following: At high temperature (\beta\leqq\beta_c) and zero field (h=0) we see a completely random state where the spins are independent and identically distributed, even independently of the type of atom (each marginal of \mu^\ox equals \lambda_o), with no preference for + or -. The state ordered by a positive field (h> 0) has a positive magnetization (expected spin \psi^{x_i}(s)=z_{x_i}>0 at each site), although the spins remain independent and there is lack of long range order in the sense that the magnetization (z_1 or z_0) at site i varies with the type (A or B) of the occupant of i. At \beta>\beta_c, h=0 we have a ferromagnetically ordered state where all the spins simultaneously obey either \mu_+^\ox or \mu_-^\ox, and in both cases they tend to align themselves, without the force of an external field. The phase transition is also reflected in the large deviations of empirical averages, and the zeroes of the rate functions indicate where the mass concentrates in the limit n\to\infty. Let I_\infty be the rate function of the laws \mu_n^\ox(\empRn\in\ett). Recall the definition \pi\mu^\piste \equiv\int \mu^\oy\pi(d\oy) from Section 2. Then we have I_\infty(Q)=0 \text{ iff } \cases Q=\pi\mu^\piste &\text{ in case\beta\leqq\beta_c$or$h\ne 0$, }\\ \text{$Q=\pi\mu_+^\piste$or$Q=\pi\mu_-^\piste$}&\text{ in case$\beta>\beta_c$and$h=0$.} \endcases \tag 4.7 $$The uniqueness/nonuniqueness dichotomy is similarly evident in the rate function S of the laws \mu_n^\ox(\frac 1n\sum_1^n\os_i \in\ett) of the average block spin. Put z(\beta,h)=pz_1(\beta,h)+(1-p)z_0(\beta,h).$$ S(t)=0 \text{ iff } \cases t=z(\beta,h) &\text{ in case$\beta\leqq\beta_c$or$h\ne 0$, }\\ \text{$t=z(\beta,0)$or$t=-z(\beta,0)$}&\text{ in case$\beta>\beta_c$and$h=0.} \endcases \tag 4.8 $$The {\it specific magnetization} is defined by$$m(\beta,h)=\lnn \,\mu_n^\ox\bigl(\sum_{i=1}^n \os_i\bigr),$$a limit which exists \pi-a.s. and is given by$$\aligned m(\beta,0)=0 &\text{ for all\beta>0$, and}\\ m(\beta,h)=z(\beta,h) &\text{ for all$\beta>0$and$h\ne 0.} \endaligned \tag 4.9 $$The spin-flip symmetry of the Hamiltonian forces \mu_n^\ox(\os_i)=0 if h=0. So in the absence of a symmetry-breaking field, the magnetization m(\beta,0) does not see the phase transition as our model does not incorporate boundary conditions. As functions of h, the quantities z_1(\beta,h), z_0(\beta,h), and m(\beta,h) are strictly increasing. Thus the limits$$m(\beta,+)=\lim_{h\searrow 0} m(\beta,h) \quad\text{ and }\quad m(\beta,-)=\lim_{h\nearrow 0} m(\beta,h)$$exist, and we shall prove that$$m(\beta,\pm)=\pm z(\beta,0). \tag 4.10 $${}From this we see that m(\beta,+)=0=m(\beta,0) if \beta\leqq\beta_c but m(\beta,+)>0=m(\beta,0) if \beta>\beta_c. Thus at low temperatures our model exhibits {\it spontaneous magnetization}, another aspect of the phase transition. Suppose for the moment that h\geqq 0. Since the A-atoms interact ferromagnetically among themselves and the B-atoms antiferromagnetically, it seems likely that the +-tendency of the A-atoms is stronger than that of the B-atoms, that is, z_1\geqq z_0. This turns out not to be necessarily the case, and even more surprisingly, the cut-off point depends only on p and J and not at all on \beta and h, provided we are in the regime of nontrivial (\ne (0,0)) solutions: \proclaim{4.11. Proposition} Suppose \beta>\beta_c and h=0, or h> 0. Then$$\aligned z_1(\beta,h)>z_0(\beta,h) &\text{ iffp\leqq 1/2$or$J<(2p-1)^{-1}$,}\\ z_1(\beta,h)=z_0(\beta,h) &\text{ iff$p> 1/2$and$J=(2p-1)^{-1}$,}\\ z_1(\beta,h) 1/2$ and $J>(2p-1)^{-1}$.}\\ \endaligned $$\endproclaim Our final result looks at the behavior of some physical quantities near the critical temperature. A basic fact is that the phase transition is continuous, in that as \beta\searrow\beta_c, m(\beta,\pm)\to 0=m(\beta_c,\pm)=m(\beta_c,0). This is a consequence of the next lemma. \proclaim{4.12. Lemma} The quantities z_0(\beta,0) and z_1(\beta,0) are continuous functions of \beta, and for \beta\geqq\beta_c strictly increasing. In particular they decrease to 0 as \beta\searrow\beta_c. \endproclaim The zero-field specific heat C_{h=0} is defined by$$C_{h=0}(\beta)=-\beta^2\, \frac {\partial u}{\partial\beta} (\beta,0),$$where the specific energy$$u(\beta,h)=\lim_{n\to\infty} \frac 1n\,\mu_n^\ox(H_n^\ox)$$exists as a \pi-a.s. constant limit. We will argue that u is a continuous function of (z_0,z_1), hence a continuous function of \beta at fixed h=0, with u(\beta,0)=0 for \beta\leqq\beta_c and u(\beta,0)<0 for \beta>\beta_c. A {\it critical exponent} c is defined as follows: We write f\sim x^c as x\searrow 0 if$$\lim_{x\searrow 0}\frac {\log f(x)}{\log x} =c.$$\proclaim{4.13. Theorem} We have the following critical behavior, for x=0,1:$$ \align z_x(\beta,0)&\sim (\beta-\beta_c)^{1/2}\quad \text{as $\beta\searrow\beta_c$, }\tag 4.14\\ z_x(\beta_c,h)&\sim h^{1/3}\quad \text{as $h\searrow 0$, and}\tag 4.15\\ \frac{\partial z_x}{\partial h}(\beta,0) &\sim |\beta-\beta_c|^{-1}\quad \text{as $\beta\to\beta_c^\pm$.}\tag 4.16\\ C_{h=0}\ &\text{has a discontinuity at $\beta=\beta_c$.} \tag 4.17 \endalign $$\endproclaim We close with some remarks before turning to the proofs. \demo{4.18. Remark} These exponents agree with the classical Curie-Weiss theory of magnetism (see Section 4-6 in \cite{T}). \enddemo \demo{4.19. Remark} In (4.16) above z_x(\beta,h) is not continuous as h passes through 0 if \beta>\beta_c, so in that case the derivative is one-sided, i.e. \partial z_x/\partial h(\beta,0^\pm). \enddemo \demo{4.20. Remark} The physical theory of spin glasses \cite{FH} leads one to study the asymptotics of the overlap \frac 1n \sum_1^n\os_i\tau_i, where \tau is an independent copy of \os, and to look for a state where the overlap has nonzero limit points while m(\beta,+)=0. But our model does not exhibit such a state, as can be verified by identifying the zeroes of the rate function of the laws \mu_n^\ox\otimes\mu_n^\ox \bigl(\frac 1n \sum_1^n\os_i\tau_i\in\ett\bigr).  \enddemo \demo{4.21. Remark} We assumed 00 throughout, but the expression (4.3) for \beta_c gives correct information also about the limiting cases. Existence of these limits is guaranteed by \partial \beta_c/\partial p<0 and \partial \beta_c/\partial J<0. \roster \item In the case p=0 (all atoms are of type B) we have an antiferromagnetic Curie-Weiss model whose only limiting Gibbs measure is easily seen to be \lambda for all temperatures. Accordingly \lim_{p\searrow 0}\beta_c=\infty. \item The opposite case p=1 is the classical Curie-Weiss model, whose \beta_c=1 \cite{E}, and we have \lim_{p\nearrow 1}\beta_c=1. \item If we set J=0 the A- and B-atoms equilibrate independently of each other, which is reflected mathematically in a decoupling of the equations (4.2). The B-atoms always choose \lambda and the A-atoms behave as a dilute Curie-Weiss model with \beta_c=p^{-1}. Correspondingly \lim_{J\searrow 0}\beta_c=p^{-1}. \endroster \enddemo \demo{4.22. Remark} The constant J influences the model in seemingly contradictory ways. On the one hand, \beta_c decays like J^{-1} as J\nearrow\infty by (4.3), so we can produce a phase transition at arbitrarily high temperatures (recall that temperature \sim\beta^{-1}) by taking J large enough. An intuitively plausible explanation is that increasing J increases the effective coupling of the A-atoms, since each AA-pair is coupled not only directly but also via each B-atom. Thus the ordering effect dominates the entropy effect already at higher temperatures. In fact (4.29) shows that for any given \beta and p, we are in the ferromagnetically ordered state whenever$$J^2> \frac{(1-\beta p)(1+\beta(1-p))} {\beta^2p(1-p)}.On the other hand, J inhibits magnetization by the external field. For a fixed \beta<\beta_c, \partial z_1/\partial h decays like J^{-1} for large J. This effect is natural since the antiferromagnetic BB-couplings resist the aligning effect of the field. \enddemo \demo{4.23. Remark} Keeping \beta fixed, we can vary p and parametrize the phase transition in terms of a critical density p_c=p_c(\beta,J) of A-atoms, so that the model is in the ferromagnetic state whenever p>p_c. (But note that \beta>1 is always required for the ferromagnetic state.) For the critical exponent one finds z_x\sim(p-p_c)^{1/2} as p\searrow p_c. We leave the details to the reader. \enddemo %\input abproof To prove our results we apply the theory of Section 2 with the following twist: Since the quenched variable \ox appears explicitly in the Hamiltonian, we must work with the skew model which includes the quenched variable as a deterministic component: The map \ox\mapsto P^\ox is defined by P^\ox=\lambda\otimes\delta_\ox, so P^\ox is a measure on \OO\times\Sigma. Define the function F for probability measures \nu on \SS\times\XX byF(\nu)=\frac{\beta}2\bigl[ -\nu(s)^2+2(1+J)\nu(s)\nu(sx) -2J\nu(sx)^2\bigr]+\beta h\nu(s),$$where we write (s,x) for a generic element of \SS\times\XX, \nu(s) for the integral \int s\,\nu(ds,dx), and similarly for \nu(sx). Then -\beta H_n^\ox(\os)=nF(\empLnbar) where \empLnbar=n^{-1}\sum_1^n\delta_{(\os_i,x_i)} is the empirical distribution of the skew model, and \gamma_n^\ox=\mu_n^\ox\otimes\delta_\ox for the measure \gamma_n^\ox defined by (2.1). To find the minimizing measures in \KK_1 we need only consider \nu with \XX-marginal \pi_o, by (5.7). Let us parametrize such a measure \nu by its conditional spin expectations:$$\aligned s_0&=\nu(s\,|\,x=0)=\text{ expected spin of a B-atom,}\\ s_1&=\nu(s\,|\,x=1)=\text{ expected spin of an A-atom.} \endaligned $$The parameter (s_0,s_1) ranges over the square [-1 ,1]\times[-1,1]. In these terms \nu(s)=ps_1+(1-p)s_0 and \nu(sx)=ps_1, so$$F(\nu)=\frac{\beta}2\bigl[ p^2s_1^2-(1-p)^2s_0^2 +2Jp(1-p)s_0s_1\bigr]+\beta h\bigl[ps_1+(1-p)s_0 \bigr].\tag 4.24$$Entropy is given by$$\aligned K(\nu)=\frac {1-p}2&[ (1-s_0)\log (1-s_0)+ (1+s_0)\log (1+s_0)]\\ +\frac p2&[ (1-s_1)\log (1-s_1)+ (1+s_1)\log (1+s_1)]. \endaligned \tag 4.25 $$Set G(\nu)=K(\nu)-F(\nu). Physically G corresponds to free energy. We wish to argue that the minimum of G is taken uniquely at (z_0(\beta,h), z_1(\beta,h)) if h\ne 0 or \beta\leqq\beta_c and at \pm(z_0(\beta,h), z_1(\beta,h)) in the case h=0, \beta>\beta_c. \partial G/\partial s_x=0 iff s_{1-x}=\xi_{1-x}(s_x) for x=0,1, where the functions \xi_x are defined in (4.2). \xi_1 is strictly increasing, so it has a well-defined inverse \eta_1 on [-1,1]: s_0=\eta_1(s_1) iff s_1=\xi_1(s_0). (The reader is advised to sketch the graphs of \xi_0 and \eta_1 in the square [-1,1]\times[-1,1].) We have \partial G/\partial s_0=0 iff s_{0}=\eta_{1}(s_1). Since \partial^2 G/\partial s_0^2>0 on [-1,1], the minima of G are necessarily on the graph s_{0}=\eta_{1}(s_1). Set g(s_1)=G(\eta_1(s_1),s_1). Differentiating gives$$g'(s_1)=\beta Jp(1-p)[\xi_0(s_1)-\eta_1(s_1)], \tag 4.26$$and we can read off the minima: {\it Case I: h=0.} Now \xi_0(0)=\eta_1(0)=0, the graph of \eta_1 is increasing and crosses the square from left to right, and \lim_{s_1\to\pm 1}\xi_0(s_1)=\pm\infty. Furthermore, on [0,1] \eta_1' is strictly decreasing but \xi_0' strictly increasing, and vice versa on [-1,0]. {}From this it follows that if$$\eta_1'(0)\leqq\xi_0'(0)\tag 4.27$$g takes its unique minimum at 0, whereas if$$\eta_1'(0)>\xi_0'(0)\tag 4.28$$g takes its minima at \pm z_1 for the unique positive number z_1 that satisfies \xi_0(z_1)=\eta_1(z_1). (4.28) is equivalent to$$\beta^2(1+J^2)p(1-p)+\beta(2p-1)-1>0.\tag 4.29$$This quadratic has a unique positive root \beta_c given by (4.3), so (4.27) and (4.28) are equivalent to \beta\leqq\beta_c and \beta>\beta_c, respectively. {\it Case II: h\ne 0.} Suppose first that h>0. Compared to case I this amounts to shifting the graph of s_0=\eta_1(s_1) up and the graph of s_0=\xi_0(s_1) down. Thus the graphs of \eta_1 and \xi_0 no longer intersect at (0,0). But there is a unique point of intersection (z_1,z_0) in (0,1)\times(0,1) where the graph of \xi_0 crosses \eta_1 from below to above, and hence z_1 is a local minimum of g. There can be other local minima with either s_0<0 or s_1<0, but it is easy to check directly from (4.24)--(4.25) that G(s_0,s_1)>G(|s_0|,|s_1|) unless both s_0>0 and s_1>0. Thus the global minimum of G is attained at (z_0,z_1). The case h<0 follows from this because K(\nu) and F_{h=0}(\nu) are invariant under the spin flip (s_0,s_1) \mapsto (-s_0,-s_1). We have found the measures in \KK_1 in terms of the conditional expected spins (z_0,z_1) specified earlier after (4.3). We emphasize again that the elements of \KK_1 are measures on \SS\times\XX, and consequently the measures j_\infty(\nu) for \nu\in\KK_1 live on \OO\times\Sigma: \roster \widestnumber\item{\beta\leqq\beta_c, h=0:} \item"{\beta\leqq\beta_c, h=0:}" \KK_1=\{\lambda_o\otimes\pi_o\} and j_\infty(\lambda_o\otimes\pi_o)=\lambda\otimes\pi. \item"{\beta>\beta_c, h=0:}" \KK_1=\{\nu_+, \nu_-\}, where \nu_\pm(ds,dx)=\psi^x_\pm(ds)\,\pi_o(dx) (recall definition (4.5)), and j_\infty(\nu_\pm) =\vPsi_\pm\equiv\pi\otimes\mu_\pm^\piste. \item"{\beta>0, \; h\ne 0:}" \KK_1=\{\nu\}, where \nu(ds,dx)=\psi^x(ds)\,\pi_o(dx) (recall definition (4.4)), and j_\infty(\nu) =\vPsi\equiv\pi\otimes\mu^\piste. \endroster The next step is to prove that the measures \zeta_n^\ox=\gamma_n^\ox(\empRnbar(\ett)) converge, where \empRnbar is the empirical field of the skew model,$$\empRnbar=\nnsum \delta_{(\theta_i\os,\theta_i\ox)}. $$Only in the case \beta>\beta_c, h=0 can we have more than one limit point. The possible limit points are t\vPsi_++(1-t)\vPsi_- for 0\leqq t\leqq1. By the spin-flip symmetry of the Hamiltonian$$\zeta_n^\ox(\os_1)=\frac 1n \gamma_n^\ox\bigl(\sum_{i=1}^n \os_i\bigr)=0$$for all n and \ox. A straightforward computation gives$$t\vPsi_+(\os_1)+(1-t)\vPsi_-(\os_1)=(2t-1)((1-p)z_0+ pz_1)$$which must equal 0, so t=1/2 since (1-p)z_0+ pz_1>0. Theorem 2.11 now yields \proclaim{4.30. Lemma} For all \ox generic for \pi,$$\lim_{n\to\infty} \zeta_n^\ox= \cases \lambda\otimes\pi &\text{ in case $h=0$ and $\beta\leqq\beta_c$, }\\ (\vPsi_++\vPsi_-)/2 &\text{ in case $h=0$ and $\beta>\beta_c$,}\\ \vPsi &\text{ in case $h\ne 0$.} \endcases $$\endproclaim \demo{Proof of Theorem 4.6} Apply Theorem 2.15, Proposition 2.17, and Lemma 4.30.\qed \enddemo \demo{Proof of (4.7) and (4.8)} \OO-marginals of the measures j_\infty(\nu), \nu\in\KK_1, specified above give the zeroes of the rate function I_\infty. The contraction Q\mapsto Q(\os_1) gives the zeroes of the rate function S. \qed \enddemo (4.9) follows for h=0 from the spin-flip symmetry and for h\ne 0 from (4.8). \proclaim{4.31. Lemma} \roster \item z_0(\beta,h) and z_1(\beta,h) are strictly increasing functions of h. \item z_0(\beta,0) and z_1(\beta,0) are strictly increasing functions of \beta\geqq\beta_c. \endroster \endproclaim \demo{Proof} (1) It suffices to consider h\geqq 0. Fix \beta, let z_x^h=z_x(\beta,h), and write \xi_x^h for the functions defined in (4.2) to specify the h. Let 0\leqq k(\eta_1^h)'(z_1^h) because g''(z_1^h)>0 (recall (4.26)), and \eta_1^h(z_1^h)=\xi_0^h(z_1^h), we have \eta_1^h(t)\leqq\xi_0^h(t). This together with (4.32) and (4.33) gives \eta_1^k(t)<\xi_0^k(t), which implies that t cannot equal z_1^k. Since this holds for all t\geqq z_1^h, it must be that z_1^k\beta_c or h> 0 implies that z_0, z_1>0. Use (4.2) and the fact that \log(1+t)/(1-t) is strictly increasing: z_1>z_0 implies$$\aligned &{\aligned \beta Jpz_1-\beta(1-p)z_0+\beta h&=\frac 12\log\frac {1+z_0}{1-z_0}\\ &< \frac 12\log\frac {1+z_1}{1-z_1}=\beta J(1-p)z_0+\beta pz_1+\beta h \endaligned}\\ \Longrightarrow \quad &\beta p(J-1)z_1<\beta(1+J)(1-p)z_0 <\beta(1+J)(1-p)z_1\\ \Longrightarrow \quad &J(2p-1)<1, \endaligned $$so either p\leqq 1/2 or J<(2p-1)^{-1}. Similar reasoning shows that z_11 and z_1=z_0 leads to J(2p-1)=1. \qed \enddemo \demo{Proof of Lemma 4.12} Since z_x(\beta,0)=0 for \beta\leqq\beta_c and the strict increasingness for \beta\geqq\beta_c was proved in Lemma 4.31, we only need to show that, if \beta\searrow\beta' or \beta\nearrow\beta', \beta'\geqq\beta_c, then \lim_{\beta\to\beta'}z_x(\beta,0)=z_x(\beta',0). But this is immediate upon passing to the limit in the equations (4.2). \qed \enddemo \demo{Proof of Theorem 4.13} To save space we only outline the arguments here and leave out the tedious calculations. To prove (4.14)--(4.15), apply the expansion$$\frac 12\log\frac{1+s}{1-s}=s+\frac{s^3}3+O(s^5) \tag 4.34$$to (4.2). To prove (4.16) differentiate$$z_x(\beta,h)=\xi_x(\beta, h, \xi_{1-x}(\beta, h, z_x(\beta,h))) \tag 4.35 $$implicitly with respect to h at h=0. (4.17) Let us first verify the remarks on specific energy stated before Theorem 4.13. Since -\beta H_n^\ox=nF(\empLnbar), specific energy is given by \$$u(\beta,h)=-\beta^{-1}\,\lim_{n\to\infty} \gamma_n^\ox(F(\empLnbar)),$$hence it is clear from the LDP's (use Theorem 2.21 and (4.8)) that$$u(\beta,h)=-\beta^{-1}\,F(z_0(\beta,h),z_1(\beta,h)).$$(In the case \beta>\beta_c, h=0 the additional fact F(z_0,z_1)=F(-z_0,-z_1) is needed, for the limit above is -\beta^{-1} \bigl[F(z_0,z_1)+F(-z_0,-z_1)\bigr]/2.) The continuity of u(\beta,0) comes from Lemma 4.12, and for \beta\leqq\beta_c, u(\beta,0)=-\beta^{-1}\,F(0,0)=0. For \beta>\beta_c and x=0 and 1, the partial derivatives \partial z_x/\partial \beta exist by the implicit function theorem applied to (4.35). Since (z_0,z_1) minimizes K-F,$$\frac{\partial F}{\partial z_x}(z_0,z_1)= \frac{\partial K}{\partial z_x}(z_0,z_1)>0,$$where the inequality follows by direct computation from (4.25) since now z_x>0. {}From all this$$\align \frac {\partial u}{\partial\beta} =\frac{\partial }{\partial \beta}(-\beta^{-1}F) &= -\beta^{-1} \frac{\partial F}{\partial z_0} \frac{\partial z_0}{\partial \beta}- \beta^{-1} \frac{\partial F}{\partial z_1} \frac{\partial z_1}{\partial \beta}\\ &= -\beta^{-1} \frac{\partial K}{\partial z_0} \frac{\partial z_0}{\partial \beta}- \beta^{-1} \frac{\partial K}{\partial z_1} \frac{\partial z_1}{\partial \beta} <0, \tag 4.36 \endalign $$where the second equality follows from (4.24) and the last inequality from Lemma 4.12. This shows that u(\beta,0)<0 for \beta>\beta_c. We now know that C_{h=0}(\beta)=0 for \beta<\beta_c. We will show that \lim_{\beta\searrow\beta_c}C_{h=0}>0 to establish the discontinuity claimed in (4.17). For \beta>\beta_c, start with (4.36) and use z_0=\xi_0(z_1) to get$$\aligned \beta^{-1} C_{h=0}&=\Biggl( \frac{\partial K}{\partial z_1}+ \frac{\partial K}{\partial z_0} \frac{\partial \xi_0}{\partial z_1}\Biggr) \frac{\partial z_1}{\partial \beta}+ \frac{\partial K}{\partial z_0} \frac{\partial \xi_0}{\partial \beta}\equiv D(\beta)+\frac{\partial K}{\partial z_0} \frac{\partial \xi_0}{\partial \beta}. \endaligned $$The last term on the right vanishes as \beta\searrow\beta_c so we ignore it. Differentiate (4.36) with respect to \beta, solve for \partial z_1/\partial \beta, and insert this in D(\beta). Do the other derivatives in D(\beta) directly from (4.2) and (4.25). Now apply (4.34) repeatedly to estimate various parts of D(\beta) to conclude that \lim_{\beta\searrow\beta_c}D(\beta)>0. \qed \enddemo \vskip .4in %\input proof1 \flushpar {\bf 5\ \ Proofs of the general theorems} \hbox{} \flushpar We start by proving the large deviation results presented at the end of Section 2. Let \EE_n denote the \sigma-field on \OO\times\Sigma generated by \BB^\OO_n and \BB^\Sigma. Define \Fi\in\statmitat {\OO\times\Sigma} by \Fi(d\os,d\oy)= P^\oy(d\os)\,\pi(d\oy), and let \Ga\in\MSOOSi have \Sigma-marginal \pi, or \Ga_\Sigma=\pi for short. Then by Theorem 2.13 in \cite{Se1}, the specific entropy$$h^\EE(\Ga\,|\,\Fi)=\lnV H_{\EE_n}(\Ga\,|\,\Fi)$$exists, and k(Q)=\inf_\Ga h^\EE(\Ga\,|\,\Fi) where the infimum is over \Ga with marginals Q and \pi. Recall the probability measures \vPsi_\nu^\piste and j_\infty(\nu) on \OO and \vPsi_\nu on \OO\times\Sigma, defined in (2.8) and on the line above. \proclaim{5.1. Lemma} K(\nu)= k(j_\infty(\nu))=h^\EE(\vPsi_\nu|\,\Fi) for \nu\in\ms such that K(\nu)<\infty. \endproclaim \demo{Proof} Since entropies of product measures are sums of the entropies of the factors, we get$$\aligned H_{\EE_n}(\vPsi_\nu\,|\,\Fi)&= \int H_{\EE_n}(\vPsi_\nu^\oy|\,P^\oy)\,\pi(d\oy) \\ &= |V_n|\int H(\psi_\nu^\oy|\,p^\oy)\,\pi(d\oy) \\ &= |V_n|\, H(\psi_\nu\,|\,\varphi). \endaligned $$The first and last equalities follow from the conditional entropy formula ((4.4.8) in \cite{DS} or (10.2) in \cite{V}). Hence$$\aligned k(j_\infty(\nu))\le h^\EE(\vPsi_\nu\,|\,\Fi)= H(\psi_\nu\,|\,\varphi)=K(\nu)\le k(j_\infty(\nu)), \endaligned $$where the last inequality follows from the contraction principle K(\nu)=\inf_{Q_0=\nu} k(Q). \qed \enddemo Order \bzd lexicographically, and let W^-=\{\ii\in\bzd:\ii<\bold0\} be the past of the origin. For \Ga\in\MSOOSi, let \Ga_0^{\os_{W^-},\oy} be a conditional distribution of the spin \os_\bold0 at the origin under \Ga, given the past \os_{W^-}=\{\os_\ii:\ii\in W^-\} and \oy. Analogously to (3.13) in \cite{F\"o}, we have the representation$$h^\EE(\Ga\,|\,\Fi)= \int H(\Ga_0^{\os_{W^-},\oy}|\,p^\oy)\, \Ga(d\os_{W^-},d\oy)$$whenever \Ga_\Sigma=\pi. \proclaim{5.2. Lemma} Suppose k(Q)<\infty. Then k(Q)>k(j_\infty(Q_0)) unless Q=j_\infty(Q_0). \endproclaim \demo{Proof} Pick \Ga\in\MSOOSi with marginals Q and \pi so that k(Q)=h^\EE(\Ga\,|\,\Fi). Let \gamma\in \msSi be the \EE_0-marginal of \Ga. Let \psi_{Q_0} be the minimizer for Q_0 in (2.5).$$\aligned k(Q)&=h^\EE(\Ga\,|\,\Fi)= \int H(\Ga_0^{\os_{W^-},\oy}|\,p^\oy)\, \Ga(d\os_{W^-},d\oy)\\ &\ge \int H\bigl(\int \Ga_0^{\os_{W^-},\oy}\,\Ga^\oy(d\os_{W^-})\, \big|\,p^\oy\bigr)\, \pi(d\oy)=\int H(\gamma^\oy|\,p^\oy)\, \pi(d\oy)=H(\gamma\,|\varphi)\\ &\ge K(Q_0)=k(j_\infty(Q_0)). \endaligned $$The first inequality above follows from convexity, the second from (2.5). Forcing the second inequality to be an equality gives \gamma=\psi_{Q_0}. Equality in the first inequality together with the strict convexity of relative entropy implies that \Ga_0^{\os_{W^-},\oy}=\gamma^\oy \Ga-a.s. In other words, \os_\bold0 is independent of the past \os_{W^-}, given \oy , and by shift-invariance this independence holds for all spins. We get \Ga^\oy(d\os)=\iootimes \psi_{Q_0}^{\theta_\ii\oy}(d\os_\ii), from which Q=\int \Ga^\oy\,\pi(d\oy)=j_\infty(Q_0). \qed \enddemo \demo{Proof of part of Lemma 2.20}$$\aligned r_\infty &= \inf_Q\{k(Q)-F^u(Q_0)\}\\ &= \inf_Q\{k(j_\infty(Q_0))-F^u(Q_0)\}\\ &= \inf_Q\{K(Q_0)-F^u(Q_0)\}=r_1. \endaligned $$\KK_\infty=j_\infty(\KK_1) is now immediate from Lemma 5.2. I_\infty is \lsc\ by its definition, so from \{I_\infty\le b\}\subset \{k \le r_1+b+c\} we see that \{I_\infty\le b\} is a closed subset of a compact set, hence itself compact. Taking b=0 shows that \KK_\infty is compact. Since j_\infty(\nu)_0=\nu, the projection Q\mapsto Q_0 is a continuous one-to-one map of the compact set \KK_\infty onto \KK_1, hence its inverse j_\infty is a homeomorphism. The remaining statement of Lemma 2.20 about \KK_\infty^\piste will be proved in Lemma 5.9 below where it will follow naturally from considerations in the skew model setting. \qed \enddemo \demo{Proof of Theorem 2.21} The inequalities follow from the LDP of Theorem 2.1 in \cite{Se1}: Apply Lemmas 2.1.7 and 2.1.8 of \cite{DS} to the upper semicontinuous functions \log \bold1_{\Ebar}(Q)+F^u(Q_0) and F^u(Q_0) and to the lower semicontinuous functions \log \bold1_{E^\circ}(Q)+F_\ell(Q_0) and F_\ell(Q_0), defined for Q\in\MOO. Use (2.6). \qed \enddemo \demo{Proof of Theorem 2.11} Since \KK_\infty=\{I_\infty=0\} is compact, the upper bound of (2.22), applied to the complements of open neighborhoods of \KK_\infty, implies that the laws \gamma_n^\ox(\empRn\in\ett) are tight. Consequently so are the measures \zeta_n^\ox=\gamma_n^\ox(\empRn(\ett)). Suppose \zeta_{n_j}^\ox\to Q as j\to\infty. By passing to a subsequence, we may assume that \gamma_{n_j}^\ox(\empR_{n_j}\in\ett)\to\varPi. By the upper bound of (2.22) \varPi(\KK_\infty)=1, and so$$Q=\int_{\KK_\infty} M\,\varPi(dM).\tag 5.3$$Let \La(d\nu)=\varPi(M_0\in d\nu) be the image of \varPi under the projection M\mapsto M_0. Since M=j_\infty(M_0) for M\in\KK_\infty, (5.3) implies (2.12). Since \empLn is the projection of \empRn,\break \gamma_{n_j}^\ox(\empL_{n_j}\in\ett)\to\La. The last statement of the theorem follows from (2.12) because the j_\infty(\nu)'s are i.i.d. under Assumption B. \qed \enddemo \demo{Proof of Remark 2.13} Since the three sequences in question are tight, it suffices to show that limit points Q, \varPi, and \La of these sequences determine each other uniquely. Q and \La are obviously determined by \varPi as its images. Conversely, since j_\infty is the inverse of the projection, \La determines \varPi through \varPi=\La(j_\infty\in\ett). Assuming \pi ergodic guarantees that all the measures \vPsi_\nu are ergodic. In particular, \KK_\infty is a set of ergodic measures. Thus (5.3) is the ergodic decomposition of Q, and Q determines \varPi by the uniqueness of the ergodic decomposition. \qed \enddemo For the remainder of this section we work under Assumption B. The first thing to check is that permuting finitely many coordinates of \ox does not affect the limiting behavior of \zeta_n^\ox. So suppose \tau is a permutation of V_k for some k, and set (\tau\ox)_\ii=\ox_{\tau(\ii)} for \ii\in V_k, (\tau\ox)_\ii=\ox_{\ii} for \ii\in V_k^c. Let f\in b\BB^\OO_m for some m. \proclaim{5.4. Lemma} \DDD{\lim_{n\to\infty} |\zeta_n^\ox(f) -\zeta_n^{\tau\ox}(f)|=0.} \endproclaim \demo{Proof} Letting \tau act on \OO the same way it acts on \Sigma, it is clear that P^{\tau\ox}=P^\ox\circ\tau^{-1}. If n>k, F(\empLn) is invariant under \tau, and consequently$$\zeta_n^{\tau\ox}(f)=P^{\tau\ox}(\empRn(f) \,e^{|V_n|\,F(\empLn)})/Z^{\tau\ox}_n =P^\ox(\empRn(f)\circ \tau\ett e^{|V_n|\,F(\empLn)})/Z^\ox_n.$$Now notice that f(\theta_\ii\tau(\os))=f(\theta_\ii\os) whenever \ii+V_m\cap V_k=\emptyset. The fraction of sites in V_n for which this condition fails vanishes as n\to\infty, hence$$\lim_{n\to\infty} \| \empRn(f)\circ\tau-\empRn(f)\|=0. \qed $$\enddemo \demo{Proof of Proposition 2.14} Without loss of generality we prove the proposition for the original sequence \{n\}. To construct g, let \{B_j\} be a countable basis for the topology of \MOO with the property that only finitely many B_j have diameter \ge\e for any fixed \e>0. Set$$g(\ox,Q)= \limsup_{j\to\infty} \bigl[ \bold1_{B_j}(Q)\ett\limsup_{n\to\infty} \bold1_{B_j}(\zeta_n^\ox)\bigr].$$Clearly g(\ox,Q) is either 0 or 1, and it is 1 precisely when there are infinitely many B_j that both contain Q and are visited infinitely often by \zeta_n^\ox. Since \text{diam B_j} \to 0, this is equivalent to saying that Q is a limit point of \{\zeta_n^\ox\}_{n=1} ^\infty. The previous lemma then implies that g(\ox,Q)=g(\tau\ox,Q) for all finite permutations \tau. To construct \Sigma_0, employ the functions$$h_j(\ox)=\lim_{n\to\infty}\biggl[ \max_{k\ge n}\,\bold1_{B_j^{(1/n)}}(\zeta_k^\ox) \biggr].$$Here B_j^{(1/n)} is a 1/n-fattening of B_j. So for \ox and \oy such that \{\zeta_{n}^\ox\}_{n=1}^\infty and \{\zeta_{n}^\oy\}_{n=1}^\infty are tight, h_j(\ox)=1 iff \{\zeta_n^\ox\}_{n=1} ^\infty has a limit point in \Bbar_j, and \{\zeta_{n}^\ox\}_{n=1}^\infty and \{\zeta_{n}^\oy\}_{n=1}^\infty have the same limit points iff h_j(\ox)=h_j(\oy) for all j. From the above lemma again we know that h_j is invariant under finite permutations. Set$$\Sigma_0=\lbrakk \ox: \text{ $h_j(\ox)= \int h_j\,d\pi$ for all $j$ }\rbrakk.$$An application of the Hewitt-Savage 0-1 law concludes the proof of the proposition. \qed \enddemo Our next goal is Theorem 2.15 about the convergence of the maps \gamma_n^\piste. First a simple general lemma about preserving tightness under averaging: Let Z and W be Polish spaces, \kappa\in\tnmitat W, and w\mapsto \rho_n^w, n=1,2,3,\ldots, a collection of measurable maps from W into \tnmitat Z. Define \mu_n\in\tnmitat {Z} by \mu_n=\int \rho_n^w\,\pi(dw). \proclaim{5.5. Lemma} If \{\rho_n^w\}_{n=1}^\infty is tight for \kappa-a.e. w, then \{\mu_n\}_{n=1}^\infty is also tight. \endproclaim \demo{Proof} Let \{z_j\} be a countable dense subset of Z, and let$$A_{k,\ell}=\bigcup_{j=1}^\ell \Bbar_{1/k}(z_j),$$where \Bbar_{1/k}(z_j) is the closed 1/k-radius ball around z_j. Let \e>0 and find \e_k>0 such that 2\sum_k\e_k<\e. Let W_0 be the set of w\in W for which \{\rho_n^w\}_{n=1}^\infty is tight. For each w\in W_0 and k find \ell(w,k) such that \inf_n\rho_n^w(A_{k,\ell(w,k)})>1-\e_k and w\mapsto\ell(w,k) is measurable. Let$$W_{k,\ell}=\{w\in W_0: \ell(w,k)=\ell\}.$$For each k find m(k) such that \kappa(\cup_{\ell=1}^{m(k)} W_{k,\ell})>1-\e_k. Then for all k and n,$$\aligned \mu_n(A_{k,m(k)})=&\sum_{\ell=1}^\infty \int_{W_{k,\ell}} \rho_n^w(A_{k,m(k)})\,\kappa(dw)\\ \ge&\sum_{\ell=1}^{m(k)} \int_{W_{k,\ell}} (1-\e_k)\,\kappa(dw)\\ \ge &(1-\e_k)^2\ge 1-2\e_k. \endaligned $$Now set A=\cap_k A_{k,m(k)}, and deduce \mu_n(A)>1-\e for all n. A\subset \cup_{j=1}^{m(k)}\Bbar_{1/k}(z_j) for all k, so A is compact by Lemma 3.1 on p. 29 of \cite{Pa}. \qed \enddemo A further useful technical twist is periodization: Given \os\in\OO, define \os^{(n)}\in\OO by \os^{(n)}_\ii=\os_\ii for \ii\in V_n and by requiring \os^{(n)}_{\ii+(2n+1)\jj} =\os^{(n)}_\ii for all \ii and \jj. Using this we form the stationary version of the empirical process:$$\empRn^s=\frac1{|V_n|}\sum_{\ii\in V_n} \delta_{\theta_\ii\os^{(n)}}.$$Under Assumption B we can perform the same operation also on quenched variables. \proclaim{5.6. Lemma} Let f\in b\BOO, g\in b\BB^\Sigma, and \ii\in\bzd. \roster \item \DDD{\int f(\theta_\ii\os^{(n)})\,P^{\oy^{(n)}}(d\os)= \int f(\os^{(n)})\,P^{\theta_\ii\oy^{(n)}}(d\os).} \item \DDD{\int g(\theta_\ii\oy^{(n)})\,\pi(d\oy)= \int g(\oy^{(n)})\,\pi(d\oy).} \item If f\in b\BB^\OO_n, then \DDD{\nVsum \gamma_n^{\theta_\ii\oy^{(n)}}(f)= \gamma_n^{\oy^{(n)}}(\empRn^s(f))}. \endroster \endproclaim \demo{Proof} (1) It suffices to consider functions of the form f(\os)=\prod_{\jj\in W} f_\jj(\os_\jj) for a finite set W. Write \ii\equiv\jj if \ii-\jj\in (2n+1)\bzd. Then$$\aligned \int f(\theta_\ii\os^{(n)})\,P^{\oy^{(n)}}(d\os)= &\int\prod_\jj f_\jj(\os^{(n)}_{\ii+\jj}) \,P^{\oy^{(n)}}(d\os)= \prod_{\kk\in V_n}\int \prod_{\jj\,:\,\jj+\ii\equiv\kk} f_\jj \,dp^{\oy_\kk}\\ =&\prod_{\bll\in V_n}\int \prod_{\jj\,:\,\jj\equiv\bll} f_\jj \,dp^{\oy^{(n)}_{\bll+\ii}}= \prod_{\bll\in V_n}\int \prod_{\jj\,:\,\jj\equiv\bll} f_\jj \,dp^{(\theta_\ii\oy^{(n)})_{\bll}}\\ = &\int f(\os^{(n)})\,P^{\theta_\ii\oy^{(n)}}(d\os). \endaligned $$(2) is a special case of (1). For (3), note that \empLn(\os)= \empLn(\os^{(n)})=\empLn(\theta_\ii\os^{(n)}) and that f(\os)=f(\osn) for f\in b\BB^\OO_n. If \gamma_n^{\oy^{(n)}}=P^{\oy^{(n)}}, then also \gamma_n^{\theta_\ii\oy^{(n)}}= P^{\theta_\ii\oy^{(n)}}, and (3) follows from (1). Otherwise use (1) to get$$\aligned \gamma_n^{\theta_\ii\oy^{(n)}}(f)= &P^{\theta_\ii\oy^{(n)}}(f\,e^{|V_n|\,F(\empLn)})/ Z_n^{\theta_\ii\oy^{(n)}}\\ =&\int f(\theta_\ii\os^{(n)})\, e^{|V_n|\,F(\empLn(\os))}\, P^{\oy^{(n)}}(d\os)/ Z_n^{\oy^{(n)}}= \int f(\theta_\ii\os^{(n)})\, \gamma_n^{\oy^{(n)}} (d\os). \endaligned $$Now average over \ii\in V_n to conclude. \qed \enddemo We now introduce the skew model associated with the setting of Assumption B: The quenched variable is adjoined to the process as a deterministic component by defining the measures \pbar^\ox=p^{\ox_\bold0} \otimes\delta_{\ox_\bold0} and \Pbar^\ox=P^\ox\otimes\delta_\ox on \SSbar=\SS\times\XX and \OObar=\OO\times\Sigma, respectively. This way we create a new setting, notationally distinguished by overbars, that satisfies the same assumptions as the original model. By Theorem 3.3 in \cite{Se1}, the entropy of \nubar\in\MSSbar is given by$$\Kbar(\nubar)=\cases H(\nubar\,|\,\varphi_o) &\text{if $\nubar$ has marginal $\pi_o$,}\\ \infty &\text{otherwise.} \endcases \tag 5.7 $$(In the setting of Section 3 in \cite{Se1}, U=\{\bold0\} because \ox\mapsto \pbar^\ox is \FSi_0-measurable by Assumption B.) Write \nubar\ssubb \SS for the \SS-marginal of a probability measure \nubar on \SSbar. We get the LDP of the original model by a contraction to the marginal \cite{V, Remark 1, p. 5}, hence by the uniqueness of the rate function$$K(\nu)=\inf_{\nubar:\nubar\ssubb \SS=\nu}\Kbar(\nubar), \tag 5.8$$which together with (5.7) justifies (2.9). Define \Fbar:\MSSbar\to [-\infty,c] by \Fbar(\nubar)=F(\nubar\ssubb \SS). Write \empLnbar and \empRnbar for the skew empirical distribution and field, so for example$$\empRnbar=\nVsum\delta_{(\theta_\ii\os, \theta_\ii\ox)}.$$Let \gabar_n^\ox be defined as in (2.1), and \zebar_n^\ox(\ett)= \gabar_n^\ox(\empRnbar(\ett)). Since \Fbar(\empLnbar) does not depend on the \Sigma-valued coordinate, we have \gabar_n^\ox= \gamma_n^\ox\otimes\delta_\ox. The set$$\KKbar_1=\{\nubar\in\MSSbar: \Kbar(\nubar)- F^u(\nubar\ssubb \SS)=\rbar_1\}$$and its infinite-volume counterpart \KKbar_\infty are defined as before, and they are related by the homeomorphism \jbar_\infty:\KKbar_1\to \KKbar_\infty. \proclaim{5.9. Lemma} \roster \item \KKbar_\infty=\{\vPsi_\nu:\nu\in\KK_1\} and the map \nu\mapsto\vPsi_\nu is a homeomorphism from \KK_1 to \KKbar_\infty. \item \KK_\infty^\piste =\{\vPsi_\nu^\piste:\nu\in\KK_1\} and the map \nu\mapsto\vPsi_\nu^\piste is a homeomorphism from \KK_1 to \KK_\infty^\piste. \item The map (\nu,\oy)\mapsto\vPsi_\nu^\oy is jointly measurable on \KK_1\times\Sigma. \endroster \endproclaim \demo{Proof} (1) Since$$\rbar_1=\inf_{\nu\in\ms}\lbrak \inf_{\nubar\ssubb \SS=\nu} \Kbar(\nubar)-F^u(\nu)\rbrak,$$it is clear from (2.9), (5.7), and the uniqueness of \psi_\nu that \rbar_1=r_1 and \KKbar_1=\mathbreak \{\psi_\nu:\nu\in\KK_1\}. Since \KK_1\ni\nu\mapsto\psi_\nu\in\KKbar_1 is the inverse of a one-to-one continuous map (the projection) on a compact set, it is a homeomorphism. By the part of Lemma 2.20 already proved, for (1) we need to show that \jbar_\infty(\psi_\nu)= \vPsi_{\nu}. But this is really only a matter of seeing through the formalities: Suppose \nubar\ssubb\XX=\pi_o. Strictly speaking the measure \psibar_{\nubar} associated to \nubar by (2.9) lives on \SSbar\times\XX=\SS\times\XX\times\XX, but by (5.7) we can neglect the unnecessary extra \XX-factor and identify \psibar_{\nubar} with \nubar itself. Under this identification the kernel \psibar_{\nubar}^x associated to \nubar becomes just the conditional distribution \nubar^{x} of \nubar on \SS, given x\in\XX, and then \jbar_\infty(\nubar)=\iootimes \nubar^{\oy_\ii} (d\os_\ii)\,\pi(d\oy). Taking \nubar=\psi_\nu then gives \jbar_\infty(\psi_\nu)= \vPsi_{\nu}. There is also an indirect argument via entropy: By Lemma 5.1, (5.7), and (3.6) in \cite{Se1},$$\kbar(\jbar_\infty(\psi_\nu))= \Kbar(\psi_\nu)=H(\psi_\nu|\,\varphi_o)= K(\nu)=h^\EE(\vPsi_\nu|\,\Fi)=\kbar (\vPsi_\nu).$$Hence by Lemma 5.2 \jbar_\infty(\psi_\nu)= \vPsi_{\nu}. (2) By Lemma 5.1 and (3.6) in \cite{Se1}, k(\pi\vPsi_\nu^\piste)= h^\EE(\pi\otimes\vPsi_\nu^\piste|\,\Fi)= \kbar(\pi\otimes\vPsi_\nu^\piste), hence it is clear that the homeomorphism \varrho^\piste\mapsto\pi\otimes\varrho^\piste from \MM_\pi^\piste into \MOOSi restricts to a homeomorphism from \KK_\infty^\piste onto \KKbar_\infty. (2) now follows from (1). (3) Let (\nubar,x)\mapsto q(\nubar,x) be a jointly measurable conditional distribution map on \MSSbar\times\XX, that is, x\mapsto q(\nubar,x) is a version of \nubar^x. (Since the conditioning \sigma-field \BXX is countably generated, q(\nubar,x) can be defined by the martingale convergence theorem.) Write \vPsi_\nu^\oy= \iootimes q(\psi_\nu,\oy_\ii) and recall that \nu\mapsto\psi_\nu is continuous. \qed \enddemo \demo{Proof of Theorem 2.15} By Theorem 2.11 applied to the skew model and Lemma 5.5, the measures \pi\zebar_n^{\,\piste} =\int \zebar_n^\oy\,\pi(d\oy) are tight. Thus the relative compactness of \{\gamma_n^\piste\}_{n=1}^\infty will follow from proving$$\lim_{n\to\infty} \biggl|\int \gamma_n^\oy(f^\oy)\, \pi(d\oy) - \int \zebar_n^\oy(f)\,\pi(d\oy) \biggr|=0$$for any f\in b\BB_m^{\OO\times\Sigma}, where the \sigma-field \BB_m^{\OO\times\Sigma} on \OObar is generated by (\os_\ii,\ox_\ii:\ii\in V_m). So fix such an f and let n>m. In the following calculation, use \gabar_n^\oy= \gamma_n^\oy\otimes\delta_\oy, the fact that \gabar_n^\oy(f)=\gabar_n^\oyn(f) by \FSi_n-measurability, Lemma 5.6(2) and (3), again \FSi_n-measurability, then the fact that$$\lim_{n\to\infty}\| \empRnbar(f)-\empRnbar^s(f) \|=0,$$and finally the definition of \zebar_n^\oy.$$\aligned &\int \gamma_n^\oy(f^\oy)\, \pi(d\oy)=\int \gabar_n^\oy(f)\, \pi(d\oy)=\int \gabar_n^\oyn(f)\, \pi(d\oy)\\ =&\int \nVsum \gabar_n^{\theta_\ii\oyn}(f)\, \pi(d\oy) = \int \gabar_n^{\oyn}(\empRnbar^s(f))\, \pi(d\oy) = \int \gabar_n^{\oy}(\empRnbar^s(f))\, \pi(d\oy) \\ =&\int \gabar_n^{\oy}(\empRnbar(f))\, \pi(d\oy)+o(1) = \int \zebar_n^{\oy}(f)\, \pi(d\oy)+o(1). \endaligned \tag 5.10 $$This proves the relative compactness of \{\gamma_n^\piste\}_{n=1}^\infty. Suppose \gamma_{n_j}^\piste\to\gamma^\piste as j\to\infty. Applying the upper bound of (2.22) to the skew model shows that \gabar_n^\ox(\empRnbar\in U^c)\to 0 for any open neighborhood U of \KKbar_\infty, hence the laws \gabar_n^\ox(\empRnbar\in\ett) are tight for \pi-a.e. \ox. An application of Lemma 5.5 and a passage to a subsequence if necessary imply that \pi\gabar_{n_j}^{\,\piste} (\empRbar_{n_j} \in\ett)\to \Pibar as j\to\infty, for some probability measure \Pibar on \MSOOSi, and by dominated convergence \Pibar(\KKbar_\infty)=1. The equality of the first and penultimate term in (5.10) gives, upon passing to the limit along \{n_j\},$$\int_\Sigma \gamma^\oy(f^\oy)\, \pi(d\oy)=\int_{\MSOOSi} \mu(f)\,\Pibar(d\mu). \tag 5.11 $$Letting \La=\lim_{j\to\infty} \pi\gamma_{n_j}^\piste(\empL_{n_j}\in \ett), Lemma 5.9(1) implies \Pibar=\La(\nu: \vPsi_\nu \in\ett). We can rewrite (5.11) as$$\int \gamma^\oy(f^\oy)\, \pi(d\oy)=\int \vPsi_\nu(f)\,\La(d\nu) =\iint \vPsi_\nu^\oy(f^\oy)\,\La(d\nu) \,\pi(d\oy). \tag 5.12 $$This tells us that \La is the unique measure that represents the exchangeable \pi\otimes\gamma^\piste as a mixture of the i.i.d. measures \vPsi_\nu over \nu\in\KK_1. This also gives (2.16), and the invariance of \gamma^\piste follows from (2.16). \qed \enddemo \demo{Proof of Proposition 2.17} The a.s. convergence \zeta_{n_j}^\ox\to Q follows from Proposition 2.14. Let \La be the measure appearing in the decomposition (2.12) of Q. Let \gamma^\piste be a limit point of \{\gamma_{n_j}^\piste\} realized along a further subsequence \{n_j'\}, with mixing measure \La' in its decomposition (2.16). Take f\in b\BB^\OO_m in (5.10) and (5.12) to see that$$\aligned \int j_\infty(\nu)\,\La(d\nu)= Q=&\lim_{j\to\infty}\pi\zeta_{n_j'}^\piste= \lim_{j\to\infty}\pi\gamma_{n_j'}^\piste =\pi\gamma^\piste\\ =&\iint \vPsi_\nu^\oy\,\La'(d\nu)\,\pi(d\oy) =\int j_\infty(\nu)\,\La'(d\nu). \endaligned The above decomposition of $Q$ is unique because the $j_\infty(\nu)$'s are i.i.d., so $\La'=\La$. The relatively compact sequence $\{\gamma_{n_j}^\piste\}$ must converge since it has a unique limit point. \qed \enddemo \vskip .4in %\input part4 %\input part5 %\input refs \Refs \refstyle{A} \widestnumber\key{DV IV} %book %\ref \key \by \book \publ \publaddr \yr \endref %paper %\ref \key \by \paper \pages \jour \yr \vol \endref \ref \key Ah \by A. Aharony \paper Tetracritical points in mixed magnetic crystals \pages 590--593\jour Phys. Rev. Lett.\yr 1975 \vol 34\endref \ref \key Al \by D. J. 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