% This is an AMS TeX file. It uses one special font, namely
% boldface calligraphics.
\input amstex
\loadbold
%%%%%%%%%%%%%%%% fonts
\font\tenbsy=cmbsy10
\font\sevenbsy=cmbsy7
\font\fivebsy=cmbsy5
\skewchar\tenbsy='60
\skewchar\sevenbsy='60
\skewchar\fivebsy='60
\newfam\bsyfam
\textfont\bsyfam=\tenbsy
\scriptfont\bsyfam=\sevenbsy
\scriptscriptfont\bsyfam=\fivebsy
\def\bcal{\fam\bsyfam}
\font\sevenmi=cmmi7
\font\fivemi=cmmi5
\font\sevenrm=cmr7
\font\sixrm=cmr6
\define\lnV{\lim_{n\to\infty}\frac 1{|V_n|}}
\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\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\qreg{quasiregular}
\define\Pqreg{$P$-quasiregular}
\define\Preg{$P$-regular}
\define\mqreg{marginally quasiregular}
\define\pqreg{$p$-quasiregular}
\define\preg{$p$-regular}
\define\ptrf{Probab. Theory Relat. 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$}}}
\define\sssubb#1{_{\lower2pt\hbox{$\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 $}}}
%%%%measures
\define\Ga{\varGamma}
\define\La{\varLambda}
\define\De{\varDelta}
\define\LeD{{\La\erot\De}}
\define\jLeD{{\La_j\erot\De_j}}
\define\Fi{\varPhi}
\define\Fio{\varphi}
\define\Psio{\psi}
\define\meas#1{\lambda\sosup {#1}}
\define\measx{\lambda\sosup {\ox}}
\define\measu{\lambda}
\define\measo#1{\lambda_o\sosup {#1}}
\define\measox{\lambda_o\sosup {\ox}}
\define\measou{\lambda_o}
\define\measLax{\lambda_\La\sosup {\ox}}
\define\measLa#1{\lambda_\La\sosup {#1}}
\define\Meas#1{\boldsymbol\lambda\sosup {#1}}
\define\Measx{\boldsymbol\lambda\sosup {\ox}}
\define\Measu{\boldsymbol\lambda}
\define\Measo#1{\boldsymbol\lambda_o\sosup {#1}}
\define\Measox{\boldsymbol\lambda_o\sosup {\ox}}
\define\Measou{\boldsymbol\lambda_o}
\define\MeasLax{\boldsymbol\lambda_\La\sosup {\ox}}
\define\MeasLau{\boldsymbol\lambda_\La}
\define\MeasLa#1{\boldsymbol\lambda_\La\sosup {#1}}
\define\Mpiessinf{\text{$\Meas \pi$-ess inf $\Fio$}}
\define\Mpiesssup{\text{$\Meas \pi$-ess sup $\Fio$}}
%%%%% 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\abar{{\overline a}}
\define\fbar{{\overline f}}
\define\fubar{{\underline f}}
\define\kbar{{\overline k}}
\define\kubar{{\underline k}}
\define\Abar{{\overline A}}
\define\Xbar{{\overline X}}
\define\Bbar{{\overline B}}
\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\nutil{{\tilde \nu}}
\define\rhotil{{\tilde \varrho}}
\define\thtil{{\tilde\theta}}
\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\Latil{{\widetilde \varLambda}}
\define\xhat{{\hat x}}
\define\OObar{{\overline \Omega}}
\define\OOhat{{\hat \Omega}}
\define\YYhat{{\hat \YY}}
\define\pibar{{\overline \pi}}
\define\Atil{{\widetilde A}}
\define\Htil{{\widetilde H}}
\define\Gtil{{\widetilde G}}
\define\KKtil{{\widetilde \KK}}
\define\htil{{\tilde h}}
\define\qtil{{\tilde q}}
\define\util{{\tilde u}}
%%%%% random objects
\define\empRn{{\bold R}_n}
\define\empRs{{\bold R}_n^s}
\define\empR{{\bold R}}
\define\empRntil{{\bold {\tilde R}_n}}
\define\empLn{{\bold L}_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\ostilde{{\tilde\os}}
\define\osLa{\os_\La}
\define\osLac{\os_{\La^c}}
\define\oo{{\omega}}
\define\oz{{\bold z}}
\define\om{{\boldsymbol\mu}}
\define\ogam{{\boldsymbol\gamma}}
\define\onu{{\boldsymbol\nu}}
\define\olam{{\boldsymbol\lambda}}
\define\oeta{{\boldsymbol\eta}}
\define\ox{{\bold x}}
\define\oy{{\bold y}}
%%%%%%%% controls
\define\piste{{\boldsymbol\cdot}}
\define\DDD{\displaystyle}
\define\iotimes{{\underset \ii\in\bZ^d \to \otimes}\,}
\define\lec{\;\le\;}
\define\gec{\;\ge\;}
\define\ec{\;=\;}
\define\erot{\smallsetminus}
\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{{\Cal B}_\XX}
\define\BSS{{\Cal B}_\SS}
%%%% bold calligraphics
\define\CCs{{\bcal C}}
\define\CCsbar{{ \overline \CCs}}
\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{{\boldsymbol X}}
\define\OO{\Omega}
\define\OOXX{\boldsymbol\Omega}
\define\OOLaXX{\OOXX_\La}
\define\XXbar{\overline\XX}
\define\OOXXbar{\overline\OOXX}
\define\OOLa{\Omega_\La}
\define\MOOXX{\tnmitat {\OOXX}}
\define\MSOOXX{\Statmitat {\OOXX}}
\define\MSpiOOXX{\Statmitatpi {\OOXX}}
\define\MpiOOXX{\tnmitatpi {\OOXX}}
\define\ms{\tnmitat {\SS}}
\define\mss{\tnmitat {\SSs}}
\define\MOO{\tnmitat {\OO}}
\define\MOOn{\tnmitat {\OO_n}}
\define\MSOO{\statmitat {\OO}}
\define\MOOLa{\tnmitat {\OOLa}}
\define\MOOLaXX{\tnmitat {\OOLaXX}}
\define\MOOnXX{\tnmitat {\OOXX_n}}
\define\mx{\tnmitat {\XX}} \define\mtx{\statmitat {\XX}}
\define\MMpipTh{\MM^{\pi,\piste}_\Theta}
\define\GGpipTh{\GG^{\pi,\piste}_\Theta}
\define\GGbpipTh{\GG^{\beta,\pi,\piste}_\Theta}
\define\GGspiTh{\GGs^\pi_\mmTh}
\define\GGsbpiTh{\GGs^{\beta,\pi}_\mmTh}
\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\mmGa{{\boldsymbol\varGamma}}
\define\mmLaexnd{{\mmLa^{e,\ox}_{n,\delta}}}
\define\mmGaexnd{{\mmGa^{e,\ox}_{n,\delta}}}
\define\Laexnd{{\La^{e,\ox}_{n,\delta}}}
\define\mmLaeynd{{\mmLa^{e,\oy}_{n,\delta}}}
\define\mmGaeynd{{\mmGa^{e,\oy}_{n,\delta}}}
\define\Laeynd{{\La^{e,\oy}_{n,\delta}}}
\define\Laepnd{{\La^{e,\piste}_{n,\delta}}}
\define\mmLabarxend{{{\overline{\boldsymbol
\varLambda}}^{\ox,e}_{n,\delta}}}
\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\bB{{\Bbb B}}
\define\bR{{\Bbb R}}
\define\bZ{{\Bbb Z}}
\define\bzd{{\Bbb Z}^d}
\define\brk{{\Bbb R}^k}
%%%%%%%%% 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 Entropy, Limit Theorems,
and Variational Principles \\ for \\
Disordered Lattice Systems
\endtitle
\leftheadtext{Timo Sepp\"al\"ainen}
\rightheadtext{Disordered Lattice Systems}
\author Timo Sepp\"al\"ainen
\endauthor
\affil Ohio State University
\endaffil
\toc
\subhead 1. Introduction\endsubhead
\subhead 2. The Setting\endsubhead
\subhead 3. Entropy\endsubhead
\subhead 4. The Finite Volume Model\endsubhead
\subhead 5. Thermodynamic Limits\endsubhead
\subhead 6. Infinite Volume Gibbs States\endsubhead
\subhead 7. Large Deviations\endsubhead
\subhead 8. Variational Principles\endsubhead
\subhead 9. Equivalence of Ensembles for Observables\endsubhead
\subhead 10. Equivalence of Ensembles for Measures\endsubhead
\endtoc
\address Department of Mathematics,
The Ohio State University, 231 West 18th Avenue,
Columbus, Ohio 43210, USA\endaddress
\email timosepp\@math.ohio-state.edu
\endemail
\date February 1993 (revised August 1994)
\enddate
\abstract
We
study infinite volume limits and Gibbs states
of disordered lattice systems with bounded and continuous
potentials. Our main tools are a generalization of
relative entropy for random reference
measures and a large deviation theory for
nonstationary independent processes. We find that
many familiar results of invariant potentials,
such as large deviation theorems, variational
principles, and equivalence of ensembles, continue to
hold for disordered models, with suitably modified
statements.
\endabstract
\keywords Disordered lattice systems, large deviations, variational
principles, entropy
\endkeywords
\subjclass 82B44 (Primary), 60F10 (Secondary)
\endsubjclass
\endtopmatter
\vfill\break
\document
\baselineskip=12pt
%\input summary
%\NoBlackBoxes
%\input sec1
\flushpar
{\bf 1. \ Introduction }
\hbox{}
\flushpar
This paper studies disordered lattice
systems, utilizing some recent large deviation theory
for nonstationary processes. Our three interrelated
goals are to establish infinite volume limit theorems,
to describe Gibbs states by variational
principles, and
to find the
natural entropy functions for these models
and study the role of entropy in the limit theorems and
variational principles.
Disordered lattice models are interacting
spin systems on an integer lattice $\bzd$
whose interaction potential is not necessarily shift invariant.
The loss of invariance is compensated by making the
interaction $\mmPhi^\ox$
dependent on an auxiliary parameter $\ox$, called
the {\it quenched variable}, in a way
that respects the $\bzd$ action: $\mmPhi^{\theta_\ii\ox}=
\mmPhi^\ox\circ\theta_{-\ii}.$ (Precise definitions
follow in section 2.) Thus the mathematical
description of the model involves two spaces
with $\bzd$ actions, the space
$\OO$ of spin
configurations and the space $\XX$ of quenched variables.
$\bzd$ acts on $\OO$ by shifts or translations and on
$\XX$ by some homeomorphisms we need not specify.
In section 2 we explain how three familiar models,
namely the random field Ising model, the
$A_pB_{1-p}$-model of a quenched mixed magnetic
crystal, and the Edwards-Anderson spin glass model,
fit into our framework.
We refer the reader to \cite{Z} for a general overview
of disordered models and for
references to past work.
As usual in equilibrium statistical mechanics, our interest
focuses on various aspects of the thermodynamic equilibrium
of the spins in the infinite volume limit. The quenched
variables do not participate in the thermodynamic
equilibrium, but they function as constraints of the
equilibrium by specifying an `environment' for the spins:
The equilibrium of the spins takes place under a fixed
quenched variable $\ox$ or under a fixed distribution $\pi$
on $\XX$. This situation is called {\it quenched disorder}.
We study the setting from two points
of view, called the {\it skew model} and the {\it
sample model}.
The skew model lives on the product space
$\OOXX:=\OO\times\XX$
and probability measures describing equilibria are joint
distributions of spins and quenched variables.
The special role of the quenched variable is reflected in
the marginal distribution on $\XX$: Either a point mass
$\delta_\ox$ or a fixed distribution $\pi$, while the
spins are either independent under an a priori measure
or governed by a Gibbs distribution.
The sample model lives on the space $\OO$,
and is got from the skew model by
projecting all the relevant functions and
measures to $\OO$.
Stating results for the skew and sample models separately may
seem like an unnecessary duplication,
but we shall find that the
large deviations and variational
principles of the two models have
interestingly differing mathematical descriptions.
We restrict to absolutely summable
interactions that are continuous functions of the
spin and the quenched variable. These are certainly
not optimal
assumptions for many of our
results.
But with this restriction we can vary the
distribution $\pi$ of the quenched variable
freely
over the space of {\it all} invariant measures on $\XX$ and treat
the thermodynamic
quantities, pressure, entropy, and energy,
as functions of $\pi$. We also get
deterministic results
for fixed quenched variables in addition to $\pi$-almost
sure results.
And local specifications are well-defined for all
configurations,
so infinite volume Gibbs states can be
defined by DLR-equations.
In contrast, other treatments typically fix
an i.i.d. distribution $\pi$ on the quenched variables
and prove results for a.e. $\ox$. The advantage of this
approach is that the interaction need not be uniformly
bounded, but only sufficiently integrable under
$\pi$.
Next a brief overview of the contents.
Section 2 describes the setting. Section 3
introduces a generalization of relative entropy
suitable for handling situations where the reference measure
is random. This entropy function is useful
for the large deviation theory of nonstationary, independent
processes,
which we summarize in section 3 for later use.
In section 4 we characterize
Gibbs states of finite volume skew and sample models
in terms of variational principles,
paving the way for the infinite volume variational principles
of section 8.
Section 5 moves from finite
to infinite volume considerations.
We prove
the existence of the thermodynamic limit of the
pressure. (The remark on p. 180 of \cite{Gr} is in
order here, namely that pressure may have to be
interpreted as free energy, depending on the model.
We follow \cite{Is} and \cite{Z}
and call this quantity pressure.)
The pressure is a function of the
interaction and the quenched variable.
Via the idea of variables generic for a
measure we can treat the pressure also
as a function of the distribution $\pi$ of the quenched variable.
Additionally, we prove the thermodynamic
limit for a function that formally assumes the role of energy
in the finite volume variational principle of the
sample model. In section 8 we see that this limit
takes on the role of specific energy in the infinite
volume variational principle.
Section 6 defines various classes of measures
and measurable families of measures on $\OOXX$ and
$\OO$ that satisfy
DLR-equations and hence are
Gibbs states of the infinite volume models.
Section 7 studies the large deviations of empirical measures
under conditioned skew Gibbs states and under sample Gibbs
states. The rate functions are expressed in terms of
entropies relative to Gibbs kernels.
In section 8 we show that infinite volume
Gibbs states satisfy variational principles.
Sections 9 and 10 apply the large deviation theory
to derive equivalence of ensembles results.
In section 9 we repeat
the results of \cite{La} for observables.
In section 10 we
look at equivalence of ensembles at the level of measures,
seeking results that parallel
those of \cite{DSZ} and \cite{Ge2} for invariant
potentials.
Throughout the paper we find that results familiar from the
invariant theory continue to hold for the skew model,
often with only small changes.
The sample model picture is not as complete:
Under large deviations and variational principles
our results cover the sample model as well as the skew model.
However, since
the Hamiltonian is a function of both the spin and the
quenched variable, it is not clear what `conditioning
on an energy surface' might mean in the sample model,
hence our equivalence of ensembles results
are worked out in the skew model setting.
The obvious shortcoming of our results is that they are
very general and soft-analytic in nature. Explicit
computations for interesting statistical mechanical
models are often much harder than
general theorems, and it remains to be seen whether our
approach can say anything interesting about particular
models. For some disorderd mean-field models large
deviation techniques can yield exact numerical results:
In \cite{Se4} this approach is used to rigorously calculate
critical exponents of a Curie-Weiss version of the
$A_pB_{1-p}$-model. A mean-field model is more amenable
to large deviation techniques because its
Hamiltonian is a function of the empirical distribution
(a `level 2' object in the Donsker-Varadhan
large deviation theory) whereas the
Hamiltonian of a Gibbsian model with an interaction
potential is a function of the
empirical field (a level 3 object), and level 2 rate
functions are much easier to calculate explicitly than
level 3 rate functions.
The use of large deviation
theory in the statistical mechanics of
invariant potentials has been well-established
since the pioneering work of Lanford
\cite{La}. Further examples of this work and references
can be found in
\cite{Com1}, \cite{DSZ}, \cite{E}, \cite{FO}, \cite{Ge2},
\cite{Ol}, \cite{Str},
and \cite{SZ}.
An early use of large deviation theory to study
spin glasses was made in \cite{HEC}. Their mean-field-type
Hamiltonian can be formulated as a function of order
parameters to take advantage of the i.i.d.
large deviation theory of the spins under the
a priori measure.
But to develop large deviation techniques for spin glasses
that yield the same results as the tools for
invariant statistical mechanics requires a large
deviation theory for nonstationary processes. Such results
were first presented in \cite{Com2}
for i.i.d. quenched variables.
The reader will find numerous physical examples
described in \cite{Com2}, whose assumptions of continuity and boundedness of
the interaction are the same as ours.
The idea of generic variables appears in
\cite{Le} in a proof of the thermodynamic limit of
a random Ising model.
A recent related paper is
\cite{Z}, which also lays down elements of a theoretical
framework for disordered lattice systems.
With techniques completely different from ours
it proves thermodynamic limits and variational
principles for
interactions that are unbounded
but integrable under the
distribution of the quenched variable. In section 8 we
connect our variational principle with that of \cite{Z}.
The emerging theory of Markov chains with random
transitions (MCRT), also called Markov chains in random environments,
deserves to be pointed out to the reader here,
though we will not pursue this
direction in the present paper.
But there is an obvious connection
between MCRT's and noninvariant interactions, just as between
time-homogeneous Markov chains and invariant potentials
(for the latter, see \cite{Ge1}).
\cite{Cog1}--\cite{Cog-3} and
\cite{Or2} have studied MCRT's and contain
numerous references
to earlier work.
A large deviation theory for MCRT's
is developed in \cite{Se3}.
\vskip .4in
%\input sec2
\flushpar
{\bf 2. \ The Setting}
\hbox{}
\flushpar
Let $\bZ^d$ be the $d$-dimensional integer lattice,
and $\OO:=\SS^{\bzd}$
a configuration space with spins in a
Polish space $\SS$. $\bzd$ carries the $\ell\sosup 1$-topology
normed by
$|\ii|:=|\ii_1|+\cdots +|\ii_d|$.
The shift maps $\theta_\ii$ on
$\OO$ are defined by $(\theta_\ii\os)_\jj=\os_{\ii+\jj}$
for $\ii,\jj\in\bzd$ and $\os\in\OO$.
For finite $\La\subset\bzd$, write
$\La\subset\subset\bzd$, and
define finite-volume configuration
spaces $\OOLa:=\SS^\La$.
$\RR$ stands for the
class of finite rectangles in $\bZ^d$.
The space
$\XX$ of quenched variables
is another Polish space, equipped
with a continuous $\bZ^d$ action: there is
a group of homeomorphisms
$ \{\theta_\ii :\ii\in {\bZ}^d\}$
on $\XX$ satisfying $\theta_\ii\circ\theta_\jj=\theta_{\ii+\jj}$.
Set $\SSs=\SS\times\XX$, $\OOXX_{\La}=\OOLa\times\XX$,
and $\OOXX=\OO\times\XX$.
The $\bZ^d$ action on
$\OOXX$ is given by $\mmtheta_\ii(\os,\ox)
=(\theta_\ii \os,\theta_\ii \ox)$.
As a rule, boldface notation will distinguish the skew
model from the sample model.
The Borel fields of $\SS$ and $\XX$ are
denoted by $\BSS$ and $\BXX$, respectively.
$\FF\ssub\La$ is the
$\sigma$-field generated by the spins $(\os_\ii:\ii\in\La)$,
and $\FFs\ssub\La:=\FF\ssub\La\vee\BXX$.
$\cbOO$ denotes the Banach space of bounded
continuous functions on $\OO$ under the supremum norm
$\|\ett\|$. $\CC_\La$ is the subspace
of $\FF_{\La}$-measurable functions in $\cbOO$. The class
$\CC$ of local functions
is the union of the $\CC_{\La}$ over
$\La\sbsb\bzd$. Similarly on $\OOXX$ we have
the classes $\CCs_{\La}$ of $\FFs_{\La}$-measurable $\cbOOXX$-functions and
their union $\CCs$. The uniform closure $\CCsbar$
of $\CCs$ is a Banach subspace of $\cbOOXX$.
Note that while $\XX$ may also be an infinite product space,
a $\CCs$-function need not be local in the $\ox$-variable.
$\ms$
denotes the space of Borel
probability measures on $\SS$.
Spaces of probability measures are
always endowed
with their weak topologies, generated by
bounded continuous functions. It is important to note
that the weak
topologies of $\MOO$ and $\MOOXX$ are also generated
by the classes $\CC$ and $\CCs$, respectively.
For each $\ox\in\XX$, assume given a probability measure
$\measox$ on $\SS$ so that the integral
$$\measox(f)=\int f\,d\measox$$
is a continuous
function of $\ox$, for each $f\in\cbs$. $\measox$
is interpreted as the a priori measure of the spin at site $\bold0$,
in the sample specified by the quenched variable $\ox$.
In view of the $\bZ^d$ action, a natural way to extend
$\measox$ to
$\OO$ is
$$\measx(d\os):={\underset \ii\in\bzd \to \otimes}
\measo {\theta_\ii \ox}(d\os_\ii),$$
so the spins are
independent under $\measx$
and the distribution
of $\os_\ii$
is $\measo {\theta_\ii \ox}$. The a priori measure on
$\OOXX$ is $\Measx:=\measx\otimes\delta_\ox$.
Restrictions to finite-volume configuration spaces
are denoted by
$$\measLax(d\osLa):={\underset \ii\in\La \to \otimes}
\measo {\theta_\ii \ox}(d\os_\ii)$$
and $\MeasLax:=\measLax\otimes\delta_\ox$.
The continuity of the integrals ensures that these
measures are continuous functions of $\ox$ in the
weak topologies. We shall leave out the superscript
`$\ox$' to indicate that the quenched variable has been
integrated out, as for example in $\Measu:=\int\Measx\,
\pi(d\ox)$.
\proclaim{2.1. Definition}
An absolutely summable, continuous,
invariant interaction potential on $\OOXX$
is a collection
$\mmPhi=\{\Fi\ssub A: A\subset\subset\bZ^d \}\subset\CCs$ of bounded local functions
satisfying
\roster
\item"(a)" Each $\Fi\ssub A$ is $\FFs\ssub A$-measurable.
\item"(b)" $\Fi\ssub A\circ \theta_\ii=\Fi\ssub {\ii+A}$
for $A\subset\subset\bZ^d$ and $\ii\in\bZ^d$.
\item"(c)" $\DDD{ \||\mmPhi \||:=\sum_{A:A\ni 0} \|\Fi\ssub A\| <\infty.}$
\endroster
Let $\bB$ denote the space of
such objects $\mmPhi$. $\bB$ is a Banach space under the norm $\||\ett\||$.
\endproclaim
Let $\La\subset\subset\bZ^d$.
The Hamiltonian
$$ H\ssub \La := \sum_{A:A\cap \La\ne\emptyset}
\Fi\ssub A
$$
is a function in $\CCsbar$.
$\mmG_{\La}$ is the probability
kernel from $(\OOXX,\FFs_{\La^c})$ into
$(\OOXX,\FFs)$ that
acts on bounded Borel functions $f$ on $\OOXX$ by
$$\mmG^{\os,\ox}_{\La}(f) :=
\frac 1{Z^{\os,\ox}_{\La}} \int_{\OOXX} f\,e^{-H_{\La}}
\, d\MeasLax\otimes\delta_{\osLac},$$
with the partition function
$$Z^{\os,\ox}_{\La} :=
\int_{\OOXX} e^{-H_{\La}}
\, d\MeasLax\otimes\delta_{\osLac}.
$$
It is clear from
the formula that
$\mmG^{\os,\ox}_{\La}$ is
a continuous measure-valued
function of $(\osLac, \ox)$.
It acts as a continuous map on probability measures
$Q\in\MOOXX$ by the
rule
$$ (\mmG_{\La} Q)(f) =\int_{\OOXX}
\mmG^{\os,\ox}_{\La}(f) \,Q(d\os,d\ox).$$
A Hamiltonian without an external condition
$\osLac$ is defined
by
$$ H^0\ssub \La := \sum_{A:A\subset\La}
\Fi\ssub A,
$$
with the corresponding
probability kernels
$$d\mmG^{0,\ox}_{\La}:=\frac 1{Z^{0,\ox}_\La}e^{-H^0_\La}\,
d\Measx.$$
$\mmG^{0}_{\La}:\ox\mapsto\mmG^{0,\ox}_{\La}$ is a
continuous map from $\XX$ into $\MOOXX$.
The sample model is obtained
by considering $\ox$-sections
of functions $f$ on $\OOXX$, defined by
$f^\ox(\os):=f(\os,\ox)$.
Thus each $\mmPhi\in\bB$ and $\ox\in\XX$
give a potential
$$\mmPhi^\ox:=\{\Fi^\ox_A: A\subset\subset\bZ^d \}$$
on $\OO$, no longer necessarily invariant. The
Hamiltonians are $H^\ox_{\La}$ and $H^{0,\ox}_{\La}$.
The partition functions are the same as for the
skew model, and the
probability kernels $G^{\os,\ox}_\La$, $\os\in\OO\cup\{0\}$,
are
the $\OO$-marginals of the corresponding skew kernels
$\mmG^{\os,\ox}_\La$.
Note that $\ox$ remains constant when the
kernels of the sample model act on measures:
$$ (G^\ox_{\La} \mu)(f) =
\int_\OO G^{\os,\ox}_{\La}(f) \,\mu(d\os)$$
for $\mu\in\MOO$ and bounded Borel functions $f$ on $\OO$.
The contribution of the spin at site $\bold0$
is measured by
$$\Fio:=\sum_{A:A\ni \bold0} \frac{\Fi\ssub A}{|A|}. \tag 2.2 $$
The assumption $\||\mmPhi \||<\infty$ guarantees that
$\Fio\in\CCsbar$.
Throughout the paper, we take
infinite volume limits along the fixed sequence of
cubes
$$V_n:=[-n,n]^d\cap\bzd.$$
To simplify
the notation, the subscript $V_n$ is replaced by $n$,
as in $H_n=H_{V_n}$.
Invariance and ergodicity under the $\bzd$ action on $\XX$
play a major role in our development. Write
$\mtx$ for the space of invariant Borel
probabilities on $\XX$.
Call a quenched variable
$\ox$ generic for $\pi$, if $\pi\in\mx$ and
$$\lim_{n\to\infty} \nVsum \delta\Ssub{\theta_\ii\ox}=\pi$$
in the weak topology of $\mx$. It follows that $\pi\in\mtx$.
Say $\ox$ is generic if the above holds for some (unspecified) $\pi$.
By the ergodic theorem,
$$\pi\{\ox: \text{ $\ox$ is generic }\}
=1$$
for all $\pi\in\mtx$.
In particular, $\pi\in\mtx$ is ergodic
if and only if
$$\pi\{\ox: \text{ $\ox$ is generic for $\pi$}\}
=1. $$
When working with a fixed $\pi\in\mx$, we write $\MpiOOXX$
for the set of probability measures on $\OOXX$ with
$\XX$-marginal $\pi$. $\MSpiOOXX:=\MpiOOXX\cap\MSOOXX$
is the subclass of invariant probability measures.
Empirical measures are defined as follows:
The empirical distributions are
$$L_n(\os)=\nVsum\delta\Ssub{\os_\ii}\qquad\text{ and }
\qquad
\empLn(\os,\ox)=\nVsum\delta\Ssub{(\os_\ii,\theta_\ii\ox)}$$
for the sample and skew models, respectively.
The empirical fields are
$$R_n(\os)=\nVsum\delta\Ssub{\theta_\ii\os}\qquad\text{ and }
\qquad
\empRn(\os,\ox)=\nVsum\delta\Ssub{\mmtheta_\ii(\os,\ox)}.$$
The results of this paper rest on a Donsker-Varadhan type
large deviation theory for independent but nonstationary
random fields.
To remind the reader, here is the conventional
form of the {\ldp}\
(see \cite{DS} or \cite{V}):
Suppose
$\xi_n$ are random variables
with values in a Polish space $\UU$. We say that
the distributions
$P\{\xi_n\in\ett\}$ on $\UU$
satisfy a \ldp\ with rate $I$ if these two requirements are met:
\roster
\item"(a)" $I$ is a lower semicontinuous
function from $\UU$ into $[0,\infty]$ with compact sublevel sets
$\{I\le c\}$ for all real $c$.
\item"(b)" For
Borel subsets $A$ of $\UU$,
$$\aligned
-\inf_{A^\circ}I&\le\lil P\{\xi_n\in A\} \\
&\le\lsl P\{\xi_n\in A\} \le
-\inf_{\Abar}I,
\endaligned
$$
where $A^\circ$ and $\Abar$ are the interior and the closure
of $A$, respectively.
\endroster
In our applications $\xi_n$ will
be an empirical measure and $\UU$ the appropriate space
of probability measures.
We conclude this section by describing how three familiar
disordered models fit the setting described here.
We give only the simplest versions of the
models with spins that take values in the
space $\SS=\{-1,+1\}$, as generalizations are
easy to write down.
\demo{2.3. Example} The
random field
Ising model (RFIM)
with nearest-neighbor
interactions can be given in terms of the
Hamiltonian
$$H_\La^{0,\ox}(\os)= -J \sum\Sb \{\ii,\,\jj\}
\subset\La\\|\ii-\jj|=1\endSb
\os_\ii\os_\jj -\sum_{\ii\in\La} h_\ii\os_\ii
\tag 2.4$$
together with the a priori measure
$$\zeta=
\bigl(\frac 12\delta_{-1}+\frac 12\delta_{+1}\bigr)
^{\otimes\bzd}.
\tag 2.5
$$
The disordered magnetic field variables
make up the quenched variable
$\ox=(h_\ii:\ii\in\bzd)$. However, requirement
(c) of the definition 2.1 of an interaction would force
us to bound the $h_\ii$ uniformly. The way around
this requirement is to subsume the disorder into
the a priori measure. Thus we shall describe
the RFIM by taking
$$\measox=\frac 1{e^{-h_\bold0}+e^{h_\bold0}}
\bigl(e^{-h_\bold0}\delta_{-1}+e^{h_\bold0}
\delta_{+1}\bigr) \tag 2.6
$$
and superimposing on $\measx$ an ordinary Ising
Hamiltonian, namely the first term of (2.4).
See \cite{AW} and \cite{BK} for recent rigorous
work on this model.
\enddemo
\demo{2.7. Example} The $A_pB_{1-p}$-model
describes the situation where 2 different types
of atoms, A and B, interact on the lattice.
The quenched variable
$\ox=(\ox_\ii:\ii\in\bzd)$ comes from setting
$\ox_\ii=1$ or $0$ according to whether
site $\ii$ is occupied by an A- or a B-atom.
The measure $\pi$ on $\XX=\{0,1\}^{\bzd}$
describes the random mechanism that determines
the occupations $\ox_\ii$. This situation
is often called {\it site disorder}. The point is
that the model can simultaneously
contain opposing types
of magnetic ordering, ferromagnetic and
antiferromagnetic, depending on the choices
of the coupling constants $J_{AA}$, $J_{BB}$, and
$J_{AB}$ for the three different possible pairs
(we are again restricting to nearest-neighbor
interactions). The Hamiltonian is given by
$$
H_\La^{0,\ox}(\os)=
-\sum\Sb \{\ii,\,\jj\}\subset\La\\|\ii-\jj|=1\endSb
\os_\ii\os_\jj
\bigl[ J_{AA}\ox_\ii \ox_\jj
+J_{BB}(1-\ox_\ii)(1-\ox_\jj)
+J_{AB}(\ox_\ii(1-\ox_\jj)+\ox_\jj(1-\ox_\ii))\bigr] $$
and the a priori measure is that of (2.5).
See \cite{Ah} for physical background.
\enddemo
\demo{2.8. Example} The Edwards-Anderson
spin glass model (EA) is an Ising model
with random coupling constants. Its Hamiltonian
is
$$H_\La^{0,\ox}(\os)= \sum\Sb \{\ii,\,\,\jj\}
\subset\La\\|\ii-\jj|\leqq R\endSb J_{\{\ii,\,\jj\}}^\ox
\os_\ii\os_\jj.
\tag 2.9
$$
For the sake of illustration, we allow
a general finite range of interaction $R\geqq 1$.
Let
$$D:=\{\ii\in\bzd: \ii> \bold0,
|\ii|\leqq R\}$$
where
$>$ on $\bzd$ denotes strict lexicographic ordering.
To adhere to
our boundedness requirement 2.1(c), we need to pick a
number $K>0$ that gives the maximum coupling
strength, $ J_{\{\ii,\,\jj\}}^\ox\leqq K$.
Let $E:=[-K,K]^D$ and $\XX:=E^{\bzd}$. A quenched
variable $\ox\in \XX$ is a doubly indexed
configuration $\ox=(\ox_{\kk,\kk+\ii}:
\kk\in\bzd, \ii\in D)$ with
$-K\leqq \ox_{\kk,\kk+\ii}\leqq K$. $\bzd$ acts
on $\XX$ by $(\theta_\jj\ox)_{\kk,\kk+\ii}=
\ox_{\kk+\jj,\kk+\ii+\jj}.$
Set
$$J_{\{\ii,\,\jj\}}^\ox:=\cases
\ox_{\ii,\,\jj} &\text{if $\jj\in\ii+D$}\\
\ox_{\jj,\ii} &\text{if $\ii\in\jj+D$}\\
0 &\text{otherwise.}
\endcases
$$
Exactly one of the cases happens for each 2-point set
$\{\ii,\,\jj\}$; this was the point of requiring
that $D$ be lexicographically strictly
above $\bold0$. On the other hand, whenever
$0<|\ii-\jj|\leqq R$, then either $\jj\in\ii+D$
or $\ii\in\jj+D$. The desired Hamiltonian (2.9)
is got by defining the interaction potential as
$$\Fi_A(\os,\ox):=
\cases J_{\{\ii,\,\jj\}}^\ox
\os_\ii\os_\jj &\text{if $A=\{\ii,\,\jj\}$ }\\
0 &\text{if $|A|\ne 2$.}
\endcases
$$
This arrangement allows nonstationary
correlations among some coupling constants:
The law of the variable $\ox_\kk=
(\ox_{\kk,\kk+\ii})_{\ii\in D}$ can be any
probability measure on $E$,
while the $E$-valued process
$(\ox_\kk)_{\kk\in\bzd}$ has to be stationary with
law $\pi$. This formulation is general enough to
contain Example 2.7 as a special case.
For general background
on the physics of spin-glasses and this model
in particular, we refer the reader to \cite{FH}.
\enddemo
\demo{2.10. Non-Example} The Sherrington-Kirkpatrick
spin glass model, with the Hamiltonian
$$H_n(\os)=\frac 1{\sqrt n}\sum_{1\leqq i0$ for all $c$, hence a well-defined inverse
$c=c(m)$ exists for $-1< m< 1$.
As probability measures on $\SS$ are uniquely
determined by expectations, $\psi_m(s)=e^{c(m)s}$.
A final thing to notice is that
$$m(c)=\frac \partial{\partial c}
\int \log(e^{c+h}+e^{-c-h})\,\pi_o(dh).
$$
Now we can compute, from (3.14), that
$$\aligned
S^\pi(\a_m)&= m\,c(m)-
\int \log(e^{c+h}+e^{-c-h})\,\pi_o(dh)+
\int \log(e^{h}+e^{-h})\,\pi_o(dh)\\
&=m\,c(m)-\int_0^c m(b)\,db\\
&=\int_0^m c(p)\,dp.
\endaligned
$$
We used the identity $m(0)=0=c(0)$ (from
the symmetry of $\pi_o$) and did an
integration by parts for the last equality.
In this example $s^\pi(\mu_m)>h(\mu_m|\,\lambda)$ for
all $m\ne -1$, $0$, $1$, so the new entropy differs from
specific relative entropy. (This is easiest to check
by first computing
$h(\mu_m|\,\lambda)=\frac 12[(1-m)\log(1-m)
+ (1+m)\log(1+m)]$ and then Taylor expanding
to see that $s^\pi(\mu_m)>h(\mu_m|\,\lambda)$
holds for small $m$; it follows for all
$m$ from convexity and the inequality
$s^\pi(\mu_m)\geqq h(\mu_m|\,\lambda)$ that is
always valid.)
\enddemo
\demo{Proof of Lemma 3.13} Set
$\nu(ds,d\ox):= \psi(s)/\measox(\psi)\,\Measou(ds,d\ox)$.
The goal is to compare the right-hand side of (3.3)
with $H(\nu\,|\,\Measou)$ which equals
the right-hand side of (3.14). We shall begin by
proving $\Hhat(\a\,|\,\measo\piste,\pi)\geqq
H(\nu\,|\,\Measou)$.
Let $\psi_b=\psi\wedge b$ for $b$ large but finite.
The hypothesis of the lemma implies that
$0<\measox(\psi_b)\leqq \measox(\psi)<\infty$ for
$\pi$-a.a. $\ox$.
Let $\log^+x=\log(x\vee 1)$ and
$\log^-x=-\log(x\wedge 1)$.
\demo{Claim 1} For all $b$, $\e>0$, we have
$\DDD{ 0\leqq \frac{\psi(s)}{\measox(\psi)}
\log^-\frac{\psi_b(s)+\e}{\measox(\psi_b)+\e}
\leqq \frac 1e\,}$.
\enddemo
Make these observations:
\roster
\item"{$(i)$}" If $0\leqq x\leqq y$
and $y>0$, then $\DDD{\frac xy
\leqq \frac {x+\e}{y+\e}}$.
\item"{$(ii)$}" If $\psi(s)\ne\psi_b(s)$ then
$\psi(s)>\psi_b(s)=b\geqq \measox(\psi_b)$ and hence
$\DDD{\log^-\frac{\psi(s)}{\measox(\psi_b)}=0}$
$\DDD{=
\log^-\frac{\psi_b(s)}{\measox(\psi_b)}}$.
\item"{$(iii)$}" The function $\log^-$ is
decreasing, hence
$\DDD{\log^-\frac{\psi(s)}{\measox(\psi_b)}\leqq
\log^-\frac{\psi(s)}{\measox(\psi)}}$.
\endroster
Now the following steps prove Claim 1:
$$\aligned
0\leqq\frac{\psi(s)}{\measox(\psi)}
\log^-\frac{\psi_b(s)+\e}{\measox(\psi_b)+\e}
&\leqq
\frac{\psi(s)}{\measox(\psi)}
\log^-\frac{\psi_b(s)}{\measox(\psi_b)}
=\frac{\psi(s)}{\measox(\psi)}
\log^-\frac{\psi(s)}{\measox(\psi_b)}\\
&\leqq \frac{\psi(s)}{\measox(\psi)}
\log^-\frac{\psi(s)}{\measox(\psi)}\leqq \frac 1e.
\endaligned
$$
\demo{Claim 2}
$\DDD{\, \liminf_{b\to\infty}\,
\lim_{\e\to 0}\, \Measou\biggl(
\frac{\psi}{\measo\piste(\psi)}
\log^+\frac{\psi_b+\e}{\measo\piste(\psi_b)+\e}\biggr)
\geqq \Measou\biggl(
\frac{\psi}{\measo\piste(\psi)}
\log^+\frac{\psi}{\measo\piste(\psi)}\biggr)
}$.
\enddemo
Fix $b$. Whenever $\psi_b(s)\geqq \measox(\psi_b)$
and $\e'<\e$, then
$\DDD{\frac{\psi_b(s)+\e}{\measox(\psi_b)+\e}
\leqq \frac{\psi_b(s)+\e'}{\measox(\psi_b)+\e'}}$.
Thus
$$
\lim_{\e\to 0} \Measou\biggl(
\frac{\psi}{\measo\piste(\psi)}
\log^+\frac{\psi_b+\e}{\measo\piste(\psi_b)+\e}\biggr)
= \Measou\biggl(
\frac{\psi}{\measo\piste(\psi)}
\log^+\frac{\psi_b}{\measo\piste(\psi_b)}\biggr)
$$
holds by monotone convegence. Claim 2 follows
as $b\nearrow\infty$ by
Fatou's lemma.
The definition (3.3) remains unchanged if the
supremum is taken over bounded Borel functions.
Thus for all positive $b$ and $\e$,
$$\aligned
\Hhat(\a\,|\,\measo\piste,\pi)&\geqq
\a(\log(\psi_b+\e))-\int \log \measox(\psi_b+\e)\,
\pi(d\ox)\\
&=\Measou\biggl(
\frac{\psi}{\measo\piste(\psi)}
\log^+\frac{\psi_b+\e}{\measo\piste(\psi_b)+\e}\biggr)
-\Measou\biggl(
\frac{\psi}{\measo\piste(\psi)}
\log^-\frac{\psi_b+\e}{\measo\piste(\psi_b)+\e}\biggr).
\endaligned
$$
Claim 1 implies that
the second term above is finite and justifies
splitting the integral in the above fashion.
Secondly, again by Claim 1, the dominated
convergence theorem applies to the second term
as we let first $\e\searrow 0$ and then
$b\nearrow\infty$. Applying Claim 2 to the first
term we get
$$
\Hhat(\a\,|\,\measo\piste,\pi)\geqq
\Measou\biggl(
\frac{\psi}{\measo\piste(\psi)}
\log^+\frac{\psi}{\measo\piste(\psi)}\biggr)
-\Measou\biggl(
\frac{\psi}{\measo\piste(\psi)}
\log^-\frac{\psi}{\measo\piste(\psi)}\biggr)
= H(\nu\,|\,\Measou).
$$
To prove the opposite inequality
$\Hhat(\a\,|\,\measo\piste,\pi)\leqq
H(\nu\,|\,\Measou)$, we may assume that
$$\infty>H(\nu\,|\,\Measou)
= \int \measox\biggl(
\frac{\psi}{\measox(\psi)}
\log\frac{\psi}{\measox(\psi)}\biggr)\,
\pi(d\ox),
$$
from which it follows that
$\measox(\psi\log\psi)<\infty$ $\pi$-a.s.
Let $f\in\cbs$ be arbitrary, and write
$$
\a(f)-\int\,\log \measox(e^f)\,\pi(d\ox)=
\int \lbrakk \frac{\measox(f\psi)}{\measox(\psi)}
-\log \measox(e^f)\rbrakk\,\pi(d\ox).
$$
For almost every $\ox$, the integrand equals
$$\aligned
&\frac{\measox(f\psi)}{\measox(\psi)}-
\frac{\measox(\psi\log\psi)}{\measox(\psi)}
+\frac{\measox(\psi\log\psi)}{\measox(\psi)}
-\log \measox(e^f)\\
=\,&\frac{\measox(\psi[f-\log \psi])}{\measox(\psi)}-
\log \measox(e^f)+
\frac{\measox(\psi\log\psi)}{\measox(\psi)}\\
\leqq\,& \log\frac{\measox(e^f)}{\measox(\psi)}
-\log \measox(e^f)+
\frac{\measox(\psi\log\psi)}{\measox(\psi)}\\
=\,&\measox\biggl(
\frac{\psi}{\measox(\psi)}
\log\frac{\psi}{\measox(\psi)}\biggr).
\endaligned
$$
This completes the proof of Lemma 3.13.
\qed
\enddemo
Before we turn to the proof of Lemma 3.15,
an auxiliary fact about entropy-minimizing measures:
\proclaim{3.19. Lemma} Suppose $\mu$ and $\nu$ are
probability measures on some measurable space $X$,
$C$ is a convex set of probability measures, $\nu\in C$,
and
$H(\nu\,|\,\mu)\leqq H(\rho\,|\,\mu)$ for all
$\rho\in C$. Then $\rho\ll\nu$ for all $\rho\in C$
such that $H(\rho\,|\,\mu)<\infty$.
\endproclaim
\demo{Proof} Write $\phi(x)=x\log x$ for short.
Suppose $\rho\in C$ is
such that $H(\rho\,|\,\mu)<\infty$ but $\rho\ll\nu$
fails. Let $A$ be a measurable set such that
$\rho(A)>0=\nu(A)$. For $0\leqq t\leqq 1$ set
$\nu_t=(1-t)\nu+t\,\rho$. In the next calculation,
note that $d\nu/d\mu=0$ on $A$, use
the convexity of $\phi$ and rearrange:
$$\aligned
H(\nu_t|\,\mu)&=\int \phi\biggl((1-t)\frac{d\nu}{d\mu}
+t\frac{d\rho}{d\mu}\biggr)\,d\mu\\
&\leqq
\int_A t\,\frac{d\rho}{d\mu}
\log\biggl(t\,\frac{d\rho}{d\mu}\biggr)\,d\mu
+ (1-t)
\int_{A^c} \phi\biggl(\frac{d\nu}{d\mu}
\biggr)\,d\mu
+ t \int_{A^c} \phi\biggl(
\frac{d\rho}{d\mu}\biggr)\,d\mu\\
&=(1-t) H(\nu\,|\,\mu)+t\bigl[ H(\rho\,|\,\mu)
+\rho(A)\,\log t\bigr].
\endaligned
$$
For small enough $t$, $H(\rho\,|\,\mu)
+\rho(A)\,\log t<0$ and
so $H(\nu_t|\,\mu)0\}\times\XX$,
and on this set
$\nu\ll\Measou\sim\measou\otimes\pi\sim
\a\otimes\pi$.
\demo{Step 3} There are measurable functions
$00\}$.
\demo{Step 4} There is a nonnegative measurable
function $\psi(s)$ such that
$\DDD{\frac{d\nu}{d\Measou}(s,\ox)=\frac
{\psi(s)}{\measox(\psi)}}$.
\enddemo
Put $\psi(s)=a(s)\,\bold1_{\{\phi>0\}}(s)$.
Then for bounded Borel functions $f$ on $\SSs$,
$$\aligned
\int f\,[\psi\otimes b]\,d\Measou&=
\int f\,\bold1_{\{\phi>0\}\times\XX}\,[a\otimes b]
\,\bold1_W\,d\Measou+
\int f\,\bold1_{\{\phi>0\}\times\XX}\,[a\otimes b]
\,\bold1_{W^c}\,d\Measou\\
&=\int f\,\bold1_{\{\phi>0\}\times\XX}\,d\nu
+
\int f\,[a\otimes b]
\,\bold1_{{W^c}\cap\{\phi>0\}\times\XX} \,d\Measou.
\endaligned
$$
Since $\nu(\{\phi>0\}\times\XX)=1$, the above line equals
$\nu(f)$ if $\Measou({W^c}\cap\{\phi>0\}\times\XX)=0$.
But this is equivalent to
$\measou\otimes\pi({W^c}\cap\{\phi>0\}\times\XX)=0$,
which in turn is equivalent to
$\a\otimes\pi({W^c})=0$ (see the proof of Step 2),
and by Step 2 this follows from $\nu(W^c)=0$.
Thus ${d\nu}/{d\Measou}=\psi\otimes b$. For Borel
subsets $B$ of $\XX$,
$$\pi(B)=\nu(\SS\times B)=
\iint_{\SS\times B}\psi\otimes b\,d\Measou
=\int_B\measox(\psi)\,b(\ox)\,\pi(d\ox),
$$
hence $b(\ox)=\measox(\psi)^{-1}$, $\pi$-a.s.
It is clear that $\psi$ is a generalized derivative
of $\a$, and so this completes the proof of Lemma 3.15.
\qed
\enddemo
\vskip .4in
%\input sec4
\flushpar
{\bf 4. \ The Finite Volume Model }
\hbox{}
\flushpar
Fix $\La\sbsb\bzd$ and $\pi\in\mx$.
For each $\ox\in\XX$, the
equilibrium state
$G_{\La}^{0,\ox}$ on $\OOLa$
defined by
$$dG_{\La}^{0,\ox}=\frac{e^{-H^{0,\ox}_\La}}
{Z^{0,\ox}_\La} d\measLax$$
is the unique probability
measure
that satisfies
the variational principle \cite{Is, p. 46}:
$$\aligned
\inf_{\a\in\MOOLa}\lbrak \a(H_\La^{0,\ox})+H(\a|\measLax)\rbrak
&=-\log Z_{\La}^{0,\ox}\\
&= G_{\La}^{0,\ox}(H_\La^{0,\ox})+H(G_{\La}^{0,\ox}|\measLax).
\endaligned \tag 4.1
$$
In this section we identify the energy and entropy functions
that permit this variational principle to hold for the
skew and sample model, while averaging
over quenched variables.
For probability measures $q$ on $\OOLaXX$, the energy in
$q$ is defined by
$$\mmU(q)=\mmU_\La(q):=q(H_\La^0),\tag 4.2$$
and we set
$$\mmS(q)=\mmS_\La^\pi(q):=\Hhat(q\,|\,
\MeasLa\piste,\pi).\tag 4.3 $$
The skew equilibrium state
is the probability measure $\nu$
on $\OOLaXX$ defined by
$$\nu(d\osLa,d\ox)
= G_{\La}^{0,\ox}(d\osLa)\,\pi(d\ox). \tag 4.4 $$
Let
$$\wp=\wp_{\La}^\pi:=\int\log Z_{\La}^{0,\ox}\,\pi(d\ox)\tag 4.5$$
be the averaged finite volume pressure.
\proclaim{4.6. Theorem} For all probability measures
$q\ne \nu$ on
$\OOLaXX$, we have
$$ \mmU(q)+\mmS(q)>-\wp=
\mmU(\nu)+\mmS(\nu).$$
\endproclaim
\demo{Proof} By convex duality and (3.3) applied
to $\mmS^\pi_\La$,
$$-\wp=\inf_{q\in\MOOLaXX}\lbrak
\mmU(q)+\mmS(q)\rbrak.$$
Integrate (4.1) against $\pi(d\ox)$
and use (3.4) to get
$-\wp= \mmU(\nu)+\mmS(\nu)$.
Let $q$ be an arbitrary measure such that
$-\wp= \mmU(q)+\mmS(q)$. Then $\mmS(q)<\infty$,
so by (3.4) $q\ssub\XX=\pi$ and
$$-\int\log Z_{\La}^{0,\ox}\,\pi(d\ox)=
\int \bigl[ q^\ox(H_\La^{0,\ox})+
H(q^\ox|\,\measLax)\bigr]\,\pi(d\ox).
$$
By the first equality of (4.1),
the integrands on the left and right
must coincide $\pi$-a.s.,
and so by the uniqueness in (4.1)
$q^\ox=G_{\La}^{0,\ox}$ $\pi$-a.s. Hence $q=\nu$.
\qed
\enddemo
Next we find the form that Theorem 4.6 takes in the sample
model.
For a probability measure $\a$ on $\OOLa$,
set
$$S(\a)=S_\La^\pi(\a):=\Hhat(\a\,|\,\measLa\piste,\pi).\tag 4.7$$
For $\a$ such that $S(\a)<\infty$,
define the quantity $U (\a)=U_{ \La}^\pi(\a)$
by the equation
$$ U (\a)+S(\a)=
\inf_{ q\ssubb {\OOLa}=\a}\lbrak \mmU(q)+\mmS(q)\rbrak.\tag 4.8$$
Let $\gamma :=\nu_{\OOLa}$ be
the $\OOLa$-marginal of $\nu$.
The next statement is immediate from Theorem 4.6.
\proclaim{4.9. Corollary} For all probability measures
$\a\ne\gamma $ on $\OOLa$,
$$ U (\a)+S(\a)>-\wp=
U (\gamma )+S(\gamma ).$$
\endproclaim
At this point the quantity
$U(\a)$ is purely an ad hoc
construction designed to reproduce the finite volume
variational principle in the sample model.
It gains some
mathematical legitimacy once we see in sections 5 and
8 that it has a well-defined infinite volume limit
that occupies
the natural role in the
infinite volume variational
principle. From (3.5)
it is immediate that
$U(\a)=\a(H^0_\La)$ whenever the Hamiltonian
$H_\La^{0,\ox}$ is independent of the quenched variable
$\ox$. Thus $U$ is a generalization of the
energy function of statistical mechanics without
disorder.
$U(\a)$ has an alternative expression as the
difference of two convex duals of pressure-type
functionals. For bounded continuous functions $f$ on
$\OOLa$, set
$$P(f):=\int\log\measLax(e^{f-H^{0,\ox}_\La})
\,\pi(d\ox),$$
its convex dual on measures defined by
$$P^*(\a):=\sup_{f\in C_b(\OOLa)}
\lbrak \a(f)-P(f)\rbrak.$$
\proclaim{4.10. Lemma}
$U(\a)=P^*(\a)-S(\a)$ whenever $S(\a)<\infty$.
\endproclaim
\demo{Proof} Extend $P$ to functions
$g\in C_b(\OOLaXX)$ by defining
$$\Pi(g):=\int\log\MeasLax(e^{g-H^0_\La})
\,\pi(d\ox).$$
The arguments used in the proof of Lemma 5.25 below
can be used to see that $\Pi$ is weakly
lower semicontinuous and convex, $\Pi=\Pi^{**}$,
and that, for
$\eta\in C_b(\OOLaXX)^*$, $\Pi^*(\eta)<\infty$
only if $\eta$ is given by a probability measure
on $\OOLaXX$ with $\XX$-marginal $\pi$. Let
$$F(\a):=\inf_{ q\ssubb {\OOLa}=\a}\Pi^*(q)$$
for measures $\a$ on $\OOLa$; it is finite only for
probability measures. From
$$q(g)\leqq \log\MeasLau(e^{g})+\| H^0_\La\| +
\Pi^*(q),$$
valid for all $g$,
we see that the sets $\{\Pi^*\leqq c\}$ are
compact. Thus $F$ is convex and lower
semicontinuous and consequenly $F=F^{**}$.
For $f\in C_b(\OOLa)$,
$$\aligned
P(f)&=\Pi(f)=\Pi^{**}(f)=
\sup_{q\in\MOOLaXX}\lbrak
q\ssub{\OOLa}(f)-\Pi^*(q)\rbrak
=\sup_{\a\in\MOOLa}\lbrak
\a(f)-F(\a)\rbrak\\
&=F^*(f).
\endaligned
$$
Then, for $\a\in\MOOLa$,
$$\aligned
P^*(\a)&=F^{**}(\a)=F(\a)=\inf_{ q\ssubb {\OOLa}=\a}
\;\sup_{g\in C_b(\OOLaXX)}\lbrak q(g)-\Pi(g)\rbrak\\
&=\inf_{ q\ssubb {\OOLa}=\a}
\;\sup_{g\in C_b(\OOLaXX)}\lbrakk q(g)-
\int\log\MeasLax(e^{g})
\,\pi(d\ox)+q(H^0_\La)\rbrakk\\
&=\inf_{ q\ssubb {\OOLa}=\a}
\lbrak \mmS(q)+\mmU(q)\rbrak\\
&= U(\a)+S(\a).\qed
\endaligned
$$
\enddemo
The measures $\nu$ and $\gamma$ also
minimize entropy under
an energy constraint.
For the skew model it is
clear from Theorem 4.6 that if
$\mmU(\nu)=c$, then $q=\nu$ is the unique
minimizer of $\mmS(q)$ subject to $\mmU(q)=c$.
The same holds also for the sample model:
\proclaim{4.11. Proposition}
Let $c$ be real and suppose that
$U (\gamma )=c$. Then
$\a=\gamma $ is the unique minimizer of $S(\a)$
subject to $U (\a)=c$.
\endproclaim
\demo{Proof} Suppose that $U (\a)=c$
and $S(\a)\leqq S(\gamma )$
for some $\a\in\MOOLa$. By the lower semicontinuity
and compact level sets of the function $ \mmU+\mmS$
on $\MOOLaXX$, there is a $q$
with $q\ssub{\OOLa}=\a$
that realizes the infimum on the
right-hand side of (4.8). Then
$$\aligned
\mmU(q)+\mmS(q)= c+ S(\a)
\leqq c +S(\gamma )= \mmU(\nu)+\mmS(\nu).
\endaligned
$$
By Theorem 4.6 $q=\nu$, and consequently
$\a=\gamma $.
\qed
\enddemo
Let us compare $\nu$ and $\gamma$ with
the equilibrium state $\mu$ under {\it annealed disorder}
where
the quenched variable $\ox$ participates
in the thermal equilibrium.
The a priori measure is now
$\MeasLau:=\int\MeasLax\,\pi(d\ox)$,
the partition function is
$$ Z_{ \La}^0:=\int_{\OOLaXX}
e^{- H_\La^0}\,d\MeasLau=\int_\XX Z^{0,\ox}_\La\,\pi(d\ox),
$$
and the equilibrium measure on $\OOLaXX$, as given
by the usual Gibbs prescription, is
$$d\mu :=
\frac1{Z_{ \La}^0} e^{- H_\La^0}\,d\MeasLau.$$
To read off the effect of annealed disorder,
let us rewrite this as
$$\aligned
\mu(d\osLa,d\ox)=\,\frac {e^{- H_\La^{0,\ox}(\osLa)}}
{Z_{ \La}^{0,\ox}}\,\measLax(d\osLa)\,
\frac {Z_{ \La}^{0,\ox}}{Z_{ \La}^{0}} \,\pi(d\ox)
=\,G_{ \La}^{0,\ox}(d\osLa)\,
\frac {Z_{ \La}^{0,\ox}}{Z_{ \La}^{0}} \,\pi(d\ox).
\endaligned
$$
Thus when we restrict to that
part of the space $\OOLaXX$ where $\ox$ is fixed,
the equilibrium of the spins $\osLa$ under $\mu$
is still
given by $G_{ \La}^{0,\ox}$.
Mathematically speaking, $G_{ \La}^{0,\ox}$ is
the conditional
distribution of $\mu $ on $\OOLa$, given $\ox$.
But the probabilities of quenched variables change
from the a priori distribution $\pi(d\ox)$ to
the equilibrium distribution
$Z_{ \La}^{0,\ox}/Z_{ \La}^{0}\, \pi(d\ox)$.
In the annealed equilibrium the spins do not obey
the sample equilibrium state
$\gamma=\int G^{0,\ox}_\La \,\pi(d\ox)$
but another mixture of the kernels $G^{0,\ox}_\La$,
namely
$$\mu_{\OOLa}=\int G^{0,\ox}_\La
\,\frac {Z_{ \La}^{0,\ox}}{Z_{ \La}^{0}}\, \pi(d\ox).$$
\vskip .4in
%\input sec5
\flushpar
{\bf 5.\ Thermodynamic Limits }
\hbox{}
\flushpar
Let $\pi\in\mtx$.
From the development in section 3 we have the
infinite volume entropies
$s^\pi(\mu)=\hhat(\mu\,|\,\meas\piste;\pi)$ for the
sample model and $\mms^\pi(Q)=\hhat(Q\,|\,\Meas\piste;\pi)$
for the skew model that satisfy
$$s^\pi(\mu)=\lnV S_n^\pi(\mu_n)=
\snLa S_\La^\pi(\mu_\La)
\tag 5.1$$
and
$$\mms^\pi(Q)=\lnV \mmS_n^\pi(Q_n)=\snLa \mmS_\La^\pi(Q_\La)
\tag 5.2$$
for all invariant measures
$\mu\in\MSOO$ and $Q\in\MSOOXX$.
This section presents the infinite volume limits
of the other two central quantities, namely energy
and pressure. For the energy of the skew model
this involves nothing but invariance:
\proclaim{5.3. Lemma} Let $Q\in\MSOOXX$. Then
$$\mmu(Q):=\lnV \mmU_n(Q_n)$$
exists and satisfies
$\mmu(Q)=Q(\Fio).$ In particular, $\mmu(Q)$ is a
continuous function of $Q$.
\endproclaim
The `energy' $U_\La^\pi$
of the sample model was defined somewhat
indirectly in terms of convex duals,
hence it is not as evident that it
should behave well under the thermodynamic limit.
We have the following result:
\proclaim{5.4. Proposition} Let $\pi\in\mtx$.
For $\mu\in\MSOO$ such that $s^\pi(\mu)<\infty$,
the thermodynamic limit
$$u^\pi(\mu):=\lnV U_{n}^\pi(\mu_n)$$
exists and satisfies
$$u^\pi(\mu)+s^\pi(\mu)=\inf_{Q_\OO=\mu}
\lbrak\mmu(Q)+\mms^\pi(Q)\rbrak.\tag 5.5$$
This defines
a Borel function $u^\pi$
on the nonempty convex subset $\{ s^\pi<\infty\}$ of
$\MSOO$, with $|u^\pi(\mu)|\leqq\|\Fio\|$.
If $\pi$ is ergodic, then $u^\pi$ is affine.
\endproclaim
The thermodynamic limit of the pressure, as a nonrandom
a.s. limit typically under i.i.d. quenched variables,
has been
proved many times over for disordered models.
Our point of view is a little different: We do not
work with a fixed
distribution on the quenched variables, but we show
that the infinite volume pressure $\wp^\ox$
is a well-defined function on the set of
all generic quenched variables, and that $\wp^\ox$
is naturally related to
an infinite volume pressure
$\wp^\pi$ defined as a function on invariant
distributions $\pi$.
\proclaim{5.6. Theorem}
\roster
\item"{$(i)$}" For each generic parameter $\ox$ there is
an infinite volume pressure $\wp^\ox$ such that
$\ox\mapsto\wp^\ox$ is
a bounded Borel function
and
$$\lim_{n\to\infty}\sup_{\os\in\OO}\bigl|
\wp^\ox-\nV\log Z_{n}^{\os,\ox}
\bigr|=0. \tag 5.7$$
\item"{$(ii)$}"
For each invariant probability distribution
$\pi$ on $\XX$ there is
an infinite volume pressure $\wp^\pi$ such that
$\pi\mapsto
\wp^\pi$
is an upper semicontinuous bounded
affine function on $\mtx$, and
$$\lim_{n\to\infty}\sup_{\os\in\OO}\bigl|
\wp^\pi-\nV\int\log Z_{n}^{\os,\ox}\,\pi(d\ox)
\bigr|=0. \tag 5.8$$
\item"{$(iii)$}"
The functions $\wp^\ox$ and $\wp^\pi$ are connected by
$$\wp^\pi=\int\wp^\ox\,\pi(d\ox),\tag 5.9$$
valid for all $\pi\in\mtx$, and
$\wp^\ox=\wp^\pi$ whenever
$\ox$ is generic for $\pi$.
In variational terms,
$$-\wp^\pi=\inf_{Q\in\MSOOXX}\lbrak \mmu(Q)
+\mms^\pi(Q)\rbrak =\inf_{\mu\in\MSOO}\lbrak
u^\pi(\mu)
+s^\pi(\mu)\rbrak. \tag 5.10 $$
\endroster
\endproclaim
Quenched randomness refers to the situation where
thermal averages are taken under a fixed
quenched variable $\ox$. From this perspective,
the interesting measures on $\XX$ are those that
can be realized as a
limit of averages
$|V_n|^{-1}\sum_{\ii\in V_n}\delta_{\theta_\ii\ox}$
for some $\ox$, in other words those that
have
generic quenched variables. As remarked earlier,
all ergodic measures belong to this class, but there can be
other measures too, depending on $\XX$ and the $\bzd$ action on it.
If $\XX$ is a configuration space
$\XX=E^{\bzd}$ and $\bzd$ acts by shifts, then every
invariant measure on $\XX$ has a generic quenched variable
by Lemma 3.1
in \cite{Se1}.
Suppose $\pi\in\mtx$ has a
generic quenched variable $\ox$ but is not ergodic,
and $\pi=\int\nu\,w(d\nu)$ is its
ergodic decomposition. Theorem 5.6
tells us that
$$\int \wp^\nu\,w(d\nu)=\wp^\pi
=\wp^\ox=\lnVl Z_n^{0,\ox}.\tag 5.11 $$
The interesting point is that the quenched variables generic
for nonergodic measures are atypical in the sense that
$$\nu\{\ox: \text{$\ox$ is generic for a nonergodic measure}\}=0$$
for any $\nu\in\mtx$, but nevertheless such a
quenched variable yields
the correct limit in (5.11).
Write $\wp^\pi(\mmPhi)$ to indicate the dependence
of the pressure on the interaction $\mmPhi$.
The familiar result about pressure-bounded linear
functionals on interactions \cite{Is, Theorem II.1.2} now takes the
following form:
\proclaim{5.12. Proposition} Let $\pi\in\mtx$ and $\xi\in\bB^*$.
Suppose there exists a constant $C$ such that
$$\xi(\mmPhi)\leqq \wp^\pi(\mmPhi)+C \tag 5.13$$
for all $\mmPhi\in\bB$. Then there is a unique
$Q\in\MSpiOOXX$ such that $\xi(\mmPhi)=-Q(\Fio)$ for all
$\mmPhi\in\bB$, where $\Fio$ is associated to $\mmPhi$ as in
{\rm (2.2)}.
\endproclaim
The rest of this section is devoted to the proofs.
For readers familiar with the use of large deviation theory
it perhaps suffices to say that Theorem 5.6
follows from the \ldp\ stated in (3.e) and some convex
analysis. Proposition 5.12
is proved as in \cite{Is} with a few
additional technicalities.
\demo{Proof of Lemma 5.3}
Note first that for any $\De\subset\La\sbsb
\bzd$,
$$\| H_\La^0-H_\La \|\leqq c_{\De,\La} \tag 5.14$$
and
$$\| H_\La^0-\sum_{\ii\in\La}\Fio\circ\theta_\ii
\|\leqq 2c_{\De,\La}, \tag 5.15$$
with
$$c_{\De,\La}:=|\La|\cdot\sum\Sb A\ni\bold0\\ A\not\subset
\De\endSb
\|\Fi_A\|\;+\;\||\mmPhi\||\cdot|\{\ii\in\La:\ii+\De\not\subset\La\}|.
\tag 5.16$$
By $Q$'s invariance, (4.2), and (5.15),
$$Q(\Fio)=Q(\empRn\Fio)=\nV\mmU_n(Q_n)+
O\biggl(\frac{c_{r,n}}{|V_n|}\biggr)
\tag 5.17$$
for any $r0$, and note that $p(\pi,-M\cdot f)\leqq 0$
to get $\eta(f)\geqq 0$. Finally, suppose $f_n\searrow 0$ pointwise
in $\CCs_k$. Apply (5.28) to $M\cdot f_n$, use the bounded convergence
theorem to let $n\to\infty$ on the right-hand side, and deduce
$\DDD{\limsup_{n\to\infty} M\,\eta(f_n)\leqq c}$.
This implies $\eta(f_n)\searrow 0$ as $n\to\infty$.
To summarize, we have
shown that $\eta$ is a pre-integral on $\CCs_k$. By the Daniell-Stone
theorem \cite{Du, 4.5.2} and by $\eta(\bold1_{\OOXX})=1$
there is a probability measure $Q_k$ on
$\FFs_k$ such that $\eta(f)=Q_k(f)$ for all $f\in\CCs_k$.
The measures $Q_k$ are consistent, so by Kolmogorov's extension
theorem they are the $\FFs_k$-marginals
of a probability measure $Q\in\MOOXX$. Since
$\eta(f)=Q(f)$ for all $f\in\CCs$, we get
$\eta(f)=Q(f)$ for all $f\in\CCsbar$ by passing to
uniform limits. $Q$ has to be invariant by (5.28)
because $p(\pi,f-f\circ\theta_\ii)=0$
for all $f$ and $\ii$.
It remains to show $p^*(\pi,\eta)=\mms^\pi(Q)$.
Let $f\in\CCs_k$.
Then $|V_n|\empRn(f)\in\CCs_{n+k}$,
so by (3.3) and $Q$'s invariance,
$$\mmS^\pi_{n+k}(Q_{n+k})
\geqq |V_n|\lbrakk Q(f)-\int p_n(\ox,f)\,\pi(d\ox)\rbrakk.$$
Divide by $|V_{n+k}|$ and let $n\to\infty$ to get
$$\mms^\pi(Q)\geqq Q(f)-p(\pi,f)= \eta(f)-p(\pi,f).$$
Since $k$ and $f\in\CCs_k$ were arbitrary,
$\mms^\pi(Q)\geqq p^*(\pi,\eta)$.
By (5.26)
$$p^*(\pi,\eta)\geqq \frac1{|V_k|} \lbrakk Q(|V_k|f)
-\int \log\Measx( e^{|V_k|f})\,\pi(d\ox)\rbrakk$$
for $f\in\CCs_k$, whence
$p^*(\pi,\eta)\geqq {|V_k|}^{-1}\mmS^\pi_k(Q_k)$,
and since $k$ was arbitrary,
$p^*(\pi,\eta)\geqq \mms^\pi(Q)$.
(b) The functional $f\mapsto p(\pi,f)$ is continuous in
the norm topology of $\CCsbar$ and convex. By
\cite{DuS, V.3.13} the sets
$\{f\in\CCsbar: p(\pi,f)\leqq b\}$ are weakly closed
for real $b$, which says that $f\mapsto p(\pi,f)$
is weakly lower semicontinuous. Thus $p(\pi,\ett)$ is equal
to its double dual \cite{ET, Proposition I.4.1}
and this, in light of part (a), is
equivalent to part (b).
(c) Compare (5.24) and part (b).
\qed
\enddemo
\demo{Proof of Theorem 5.6}
Define
$\wp^\ox:=p(\ox,-\Fio) $
and
$\wp^\pi:=p(\pi,-\Fio)$. Boundedness of $\wp^\ox$
and $\wp^\pi$ is trivial.
$(i)$ By (5.14)--(5.15),
$$\sup_{\ox\in\XX}\,\sup_{\os\in\OO}
\bigl| p_n(\ox,-\Fio)
- \nVl Z_{n}^{\os,\ox}\bigr|\leqq 3
\frac {c_{r,n}}{|V_n|}.
\tag 5.29$$
Letting first $n\to\infty$ and then $r\to\infty$ gives
(5.7) and proves part $(i)$.
$(iii)$ (5.9) holds by definition, and
Lemma 5.25(c) gives the statement about $\wp^\ox=
\wp^\pi$. (5.5) and Lemma 5.25(b) give (5.10).
$(ii)$ Integrating over (5.29) gives (5.8).
Affinity of $\pi\mapsto \wp^\pi$ is trivial. It
remains to prove its upper semicontinuity.
\demo{Claim} If $\La\sbsb\bzd$,
$b$ is a real number and $A$ is
a compact subset of $\mx$, then the set
$\DDD{B:=\bigcup_{\pi\in A}\{q\in\MOOLaXX:
\mmS^\pi_\La(q)\leqq b\}}$
is compact.
\enddemo
To prove the Claim, let $\{q_n\}$ be a sequence in
$B$, with $\pi_n\in A$ so that
$\mmS^{\pi_n}_\La(q_n)\leqq b$. By passing to
a subsequence we may assume $\pi_n\to\pi$.
Let $\mu_n:= \measLax(d\os_\La)\,\pi_n(d\ox)$, so that
$\mu_n\to \mu:= \measLax(d\os_\La)\,\pi(d\ox)$.
Let $M>0$ and pick a
compact subset $K$ of $\OOLaXX$ such that
$\mu_n(K^c)\ell$ and an $\eta>0$
such that
$$H(P_k\,|\,(\mmG_\ell P)_k)\geqq\eta. \tag 8.3$$
By the monotonicity of relative entropy, we may pick
$k$ large enough so that $b_{\ell,k}<\eta/8$.
Let $m$ be arbitrary, and pick $n=n_m$ so that $V_n$ is the
disjoint union of the cubes
$\ii_1+V_k$, $\ldots$,$\ii_{m^d}+V_k$
for some set $\{\ii_1,\ldots,\ii_{m^d}\}\sbsb\bzd$.
For $j=1,\ldots,m^d$ set
$\De_{j}:=\ii_j+V_\ell$
and
$$\La_j:=(\ii_1+V_k)\cup\dots\cup(\ii_{j}+V_k).$$
Use first \cite{DS, (4.4.8)} and the
fact that $P_{\La_j\erot\De_j}=(\mmG_{\De_j}P)
_{\La_j\erot\De_j}$, then the monotonicity of relative entropy,
and finally (8.3) shifted by $\ii_j$ to get
$$\aligned
\int H( P^\jLeD_{\La_j}|\,(\mmG_{\De_j}P)^\jLeD_{\La_j})\,dP
&= H(P_{\La_j}|\,(\mmG_{\De_j}P)_{\La_j})\\
&\geqq H(P_{\ii_j+V_k}|\,(\mmG_{\De_j}P)_{\ii_j+V_k})\\
&\geqq\eta.
\endaligned
$$
Set $\Ga=V_n$ and use (8.2),
the choice of $k$, \cite{DS, (4.4.8)}, and the monotonicity
of entropy again to get
$$\aligned
\eta/2 &\leqq
\int H( P^\jLeD_{\La_j}|\,Q^\jLeD_{\La_j})\,dP\\
&= H(P_{\La_j}|\,Q_{\La_j})- H(P_{\jLeD}|\,Q_{\jLeD})\\
&\leqq H(P_{\La_j}|\,Q_{\La_j})- H(P_{\La_{j-1}}|\,Q_{\La_{j-1}}),
\endaligned
$$
with the very last term missing for $j=1$. Add over
$j=1,\ldots,m^d$ and use (3.a) to get
$$\eta m^d/2\leqq H(P_n\,|\,Q_n)\leqq \mmK^\pi_n(P_n).$$
Divide by $|V_n|$ and let $m\nearrow\infty$ to get
$\eta(2k+1)^{-d}/2\leqq \mmk^\pi(P)$,
contradicting $(ii)$.
\qed
\enddemo
Next we project this result to the sample model.
\proclaim{8.4. Theorem} Let $\pi\in\mtx$ and $\mu\in\MSOO$.
The following are equivalent:
\roster
\item"$(i)$" $\mu\in\GG^\pi_\Theta$.
\item"$(ii)$" $k^\pi(\mu)=0.$
\item"$(iii)$" $\DDD{u^\pi(\mu)+s^\pi(\mu)=-\wp^\pi
=\inf_{\nu\in\MSOO}\{u^\pi(\nu)+s^\pi(\nu)\}.}$
\endroster
\endproclaim
\demo{Proof} $(i)\Longleftrightarrow(ii)$ follows from
the definition of $\GG^\pi_\Theta$,
(7.14), and $(i)\Longleftrightarrow(ii)$ in
Theorem 8.1. $(ii)\Longleftrightarrow(iii)$ follows from
(5.10) and (7.13).
\qed
\enddemo
The sample \ldp\ 7.16 and variational principle 8.4
reveal
the relevance of the class $\GG^\pi_\Theta$ for
quenched randomness: If $\gamma\in\GG^\ox$ for
a parameter $\ox$ generic for $\pi$, the weak limits
of $R_n$ under $\gamma$ are contained in $\GG^\pi_\Theta$,
in the sense that for any open neighborhood of
the convex, compact set $\GG^\pi_\Theta$,
$\gamma\{R_n\in V^c\}\to 0$
exponentially fast as $n\to\infty$. In particular, if
$\GG^\pi_\Theta=\{\mu\}$, then $R_n\to\mu$
$\gamma$-a.s.
To connect our results with those of \cite{Z}, let us
look at variational principles in terms
of invariant families as defined in section 6. For the remainder of this section,
$\pi$ is a fixed {\it ergodic} element of $\mtx$,
and $Q\in\MSpiOOXX$ and $q^\piste\in\MM^{\pi,\piste}_\Theta$
are associated as in Lemma 6.3$(ii)$.
\proclaim{8.5. Lemma} For any
$\varrho^\piste\in\MM^{\pi,\piste}_\Theta$,
the entropy
$$h(\varrho^\piste|\,\lambda^\piste):=\lnV H(\varrho^\ox_n\,|\,\measx_n)
\tag 8.6$$
relative to the a priori family $\lambda^\piste$
exists as a $\pi$-a.s. limit, independent of $\ox$. If the limit
is finite, we have convergence in $L^1(\pi)$ too.
Moreover,
$$h(q^\piste|\,\lambda^\piste)=\mmh(Q\,|\,\Measu)=\mms^\pi(Q).
\tag 8.7$$
\endproclaim
\demo{Proof} (8.6) follows from subadditive ergodic theory,
see \cite{Z, Proposition 4.2} or \cite{Se1, Theorem 3.10}. Note in
particular that a nonnegative superadditive process,
integrable or not,
converges a.s. \cite{Kr, Theorem 5.4}.
To get (8.7), integrate (8.6) against $\pi$ and use
\cite{DS, (4.4.8)} and (3.10).
\qed
\enddemo
\demo{8.8. Remark} The quantity $h(\varrho^\piste|\,\lambda^\piste)$
is the negative of the entropy defined in (4.11) of
\cite{Z}.
\enddemo
Next specific energy for invariant families
(see Proposition 4.5 in \cite{Z }):
\proclaim{8.9. Lemma} For
$\varrho^\piste\in\MM^{\pi,\piste}_\Theta$,
$$u(\varrho^\piste):=\lnV \varrho^\ox(H^{0,\ox}_n) \tag 8.10$$
exists as a $\pi$-a.s. and $L^1(\pi)$ limit, and
satisfies
$u(\varrho^\piste)=
\int \varrho^\ox(\Fio^\ox)\,\pi(d\ox)$.
In particular, $u(q^\piste)=\mmu(Q)$.
\endproclaim
\demo{Proof} Use (5.15) to write
$$\nV\varrho^\ox(H^{0,\ox}_n)=\nVsum \varrho^{\theta_\ii\ox}
(\Fio^{\theta_\ii\ox}) +
O\biggl(\frac{c_{r,n}}{|V_n|}\biggr),$$
then apply the multiparameter ergodic theorem and the argument
following (5.17).
\qed
\enddemo
The last ingredient of the variational principle
is entropy relative to a Gibbs family:
\proclaim{8.11. Lemma} For
$\varrho^\piste\in\MM^{\pi,\piste}_\Theta$
and $\gamma^\piste\in\GG^{\pi,\piste}_\Theta$, the entropy
$$h(\varrho^\piste|\,\gamma^\piste):=\lnV H(\varrho^\ox_n\,|\,
\gamma^\ox_n)
$$
exists as a $\pi$-a.s. limit, independent of $\ox$, and
is given by
$$h(\varrho^\piste|\,\gamma^\piste)=h(\varrho^\piste|\,\lambda^\piste)
+u(\varrho^\piste)+\wp^\pi.$$
Also, $h(q^\piste|\,\gamma^\piste)=\mmk^\pi(Q)$.
\endproclaim
\demo{Proof} Since
$\gamma^\ox=\int G^{\os,\ox}_n\,\gamma^\ox(d\os)$
and
$G^{\os,\ox}_n=G^{0,\ox}_n\ett e^{O(c_{r,n})}$,
we have
$$\aligned
H(\varrho^\ox_n\,|\,\gamma^\ox_n)&=
H(\varrho^\ox_n\,|\,G^{0,\ox}_n)+O(c_{r,n})\\
&=H(\varrho^\ox_n\,|\,\measx_n)
+\varrho^\ox(H^{0,\ox}_n)+\log Z^{0,\ox}_n+O(c_{r,n}),
\endaligned$$
with $c_{r,n}$ again defined by (5.16).
Divide by $|V_n|$ and let
$n\to\infty$,
note that $\pi$-a.e. $\ox$ is generic
for $\pi$, and then let $r\to\infty$.
\qed
\enddemo
And now the variational principle. The
equivalence $(i)\Longleftrightarrow(iii)$ below
was deduced
in Theorem 4.8 of \cite{Z} for a class of
unbounded interactions
with a different definition of an equilibrium family.
\proclaim{8.12. Theorem} Let $\pi\in\mtx$
be ergodic, $\varrho^\piste\in\MM^{\pi,\piste}_\Theta$,
and $\gamma^\piste\in\GG^{\pi,\piste}_\Theta$.
The following are equivalent:
\roster
\item"$(i)$" $\varrho^\piste\in
\GG^{\pi,\piste}_\Theta$.
\item"$(ii)$" $h(\varrho^\piste|\,\gamma^\piste)=0.$
\item"$(iii)$" $\DDD{u(\varrho^\piste)+
h(\varrho^\piste|\,\lambda^\piste) =-\wp^\pi
=\inf_{\nu^\piste\in\MM^{\pi,\piste}_\Theta}
\{u(\nu^\piste)+
h(\nu^\piste\,|\,\lambda^\piste) \}.}$
\endroster
\endproclaim
\demo{Proof}Theorem 8.1 and the correspondence
$\MM^{\pi,\piste}_\Theta \leftrightarrow\MSpiOOXX$ of Lemma 6.3$(ii)$.
\qed
\enddemo
\vskip .4in
%\input sec9
\flushpar
{\bf 9. \ Equivalence of Ensembles for
Observables}
\hbox{}
\flushpar
Our next goal is the Lanford theory \cite{La}
of large deviations
and equivalence of ensembles for observables.
Let $\pi\in\mtx$ be fixed throughout the section.
\proclaim{9.1. Definition} A collection
$\mmg=\{g_\La: \La\sbsb\bzd\}$
of continuous functions from $\OOXX$ into $\brk$
is an {\rm observable} if for some bounded continuous function
$g$ from $\OOXX$ into $\brk$,
$$\|g_\La-\sum_{\ii\in\La} g\circ\theta_\ii\|=o(|\La|)\quad
\text{ as $|\La|\to\infty$.}
\tag 9.2$$
\endproclaim
In particular,
$\mmg=\{g_\La=(g^1_\La,\ldots,g^k_\La): \La\sbsb\bzd\}$
is an observable
if there are
interactions $\mmPsi^j\in\bB$ such that
$$g^j_\La=\sum_{A\subset\La}\Psi^j_A$$
for $j=1,\ldots,k$.
For we can take $g=(\Psio^1,\ldots,\Psio^k)$, where $\Psio^j$ is
associated to $\mmPsi^j$ as in (2.2), and then (9.2)
follows from (5.15).
To introduce an inverse temperature $\beta$ we
replace the interaction $\mmPhi$
by $\beta \mmPhi$. $\beta$ may be any real number.
The
dependence on $\beta$ is indicated by a superscript:
The skew Gibbs kernels $\mmG^{\beta,\os,\ox}_\La$ are
defined by
$$d \mmG^{\beta,\os,\ox}_\La= \frac 1{Z^{\beta,\os,\ox}_\La}
\,e^{-\beta H_\La}\,d\MeasLax\otimes\delta_{\osLac}$$
with the obvious $Z^{\beta,\os,\ox}_\La$,
the infinite volume pressure is
$$\wp^{\beta,\pi}=\lnV\int\log Z^{\beta,\os,\ox}_n\,\pi(d\ox),$$
and $\GGsbpiTh$ denotes the
class of skew Gibbs measures for
the interaction $\beta\mmPhi$ as defined in 6.2.
Let $\mmg$ be any observable and $g$ the function
associated to it by (9.2).
We begin by recording the large deviations of
the observable under the a priori measure
$\Measx$, the microcanonical
measures $\mmLa^{e,\ox}_{n,\delta}$, and
the
canonical measures
$\mmG^{\beta,\os,\ox}_n$.
Microcanonical
measures are defined by conditioning the a priori
measure on
a thin energy shell: For $e$ real, $\delta>0$,
and a Borel subset $A$ of $\OOXX$,
$$\mmLa^{e,\ox}_{n,\delta}(A):=\Measx\lbrak A\,\big|\;
|\,e-{|V_n|}^{-1}{H^0_n}\,|\leqq\delta \rbrak,
\tag 9.3$$
whenever
$$\Measx\lbrak\,
|\,e-{|V_n|}^{-1}{H^0_n}\,|\leqq\delta \,\rbrak>0;
\tag 9.4 $$
otherwise set $\mmLa^{e,\ox}_{n,\delta}:=\Measx$.
Note that by (5.15) we could just as well have defined the microcanonical
probabilities by conditioning on the events
$\{|\empRn(\Fio)-e|\leqq\delta\}$
as in \cite{DSZ},
without affecting the limiting behavior as first $n\to\infty$
and then $\delta\to 0$.
Next the rate functions:
Define
the function $I_g:\brk\to[0,\infty]$ by
$$I_g(v):=\inf_{Q(g)=v}\mms^\pi(Q). \tag 9.5$$
For $v\in\brk$, $\beta\in\bR$,
and $e\in\bR$ such that $I_\Fio(e)<\infty$, define
(imitating \cite{La}'s notation)
$$\eta^{mc}(v|e):=I_{(g,\Fio)}(v,e)-I_\Fio(e) \tag 9.6$$
and
$$\eta^c(v|\beta):=\inf_{Q(g)=v}\{ \beta\,Q(\Fio)+
\mms^\pi(Q)\}+\wp^{\beta,\pi}.
\tag 9.7
$$
For other values of $e$ set $\eta^{mc}(v|e)=\infty$.
(9.6) makes sense because
$I_{(g,\Fio)}(v,e)\geqq I_\Fio(e)$ by (9.5).
Recall also the
shorthand $g_n=g_{V_n}$.
\proclaim{9.8. Theorem} Suppose $\ox$ is generic for $\pi$.
We have the following {\ldp}s for the observable:
\roster
\item"$(i)$" The distributions
$\Measx\lbrak |V_n|^{-1}g_n \in\ett\rbrak$
on $\brk$ satisfy a \ldp\ with rate $I_g$.
\pikkuhyppy
\item"$(ii)$" Suppose $e$ is such that
$I_\Fio(e)<\infty$. Then for any fixed $\delta>0$,
{\rm(9.4)} holds for large enough $n$.
For Borel sets $A\in\BB_{\brk}$,
$$\aligned
-\inf_{v\in A^\circ}\eta^{mc}(v|e)
&\leqq \liminf_{\delta\to 0} \lil \mmLa^{e,\ox}_{n,\delta}
\lbrakk\frac {g_n}{|V_n|}\in A\rbrakk\\
&\leqq \limsup_{\delta\to 0}\lsl \mmLa^{e,\ox}_{n,\delta}
\lbrakk\frac {g_n}{|V_n|}\in A\rbrakk\\
&\leqq -\inf_{v\in\Abar}\eta^{mc}(v|e).
\endaligned \tag 9.9
$$
\pikkuhyppy
\item"$(iii)$" The distributions
$\mmG^{\beta,\os,\ox}_n\lbrak |V_n|^{-1}g_n \in\ett\rbrak$
on $\brk$ satisfy a \ldp\ with rate $\eta^c(\ett|\beta)$,
uniformly in $\os$ as in {\rm (7.9)}.
\endroster
\endproclaim
Proof of the theorem and other
claims follow at the end
of the section.
It is clear from the variational principle 8.1
and (9.7) that
$$\{v:\eta^c(v|\beta)=0\}=\{P(g):
P\in\GGs^{\beta,\pi}_\mmTh\}.$$
Hence by part $(iii)$ above,
the possible limit points of $g_n/|V_n|$ under
$\mmG^{\beta,\os,\ox}_n$ are the expectations of $g$
under Gibbs measures. More precisely, if $V$ is an
open neighborhood of the convex, compact set
$\{P(g): P\in\GGs^{\beta,\pi}_\mmTh\}$ in $\brk$, then
$
\mmG^{\beta,\os,\ox}_n
\lbrak |V_n|^{-1}g_n \in V^c
\rbrak \to 0$
exponentially fast
as $n\to\infty$.
Let us take a closer look at the important special
case $\mmg=\Phi$, our fixed interaction.
Then the rate function
$I_\Fio$ is in duality with pressure:
$$\aligned
-\wp^{\beta,\pi}=\inf_{e\in\bR}
\{\beta e+I_\Fio(e)\}
\endaligned
\tag 9.10$$
and
$$\aligned
-I_\Fio(e)=\inf_{\beta\in\bR}
\{\beta e +\wp^{\beta,\pi}\}.
\endaligned
\tag 9.11$$
A very precise description of
the correspondence
between the dual variables $\beta$ (inverse temperature)
and $e$ (energy) can be given.
Our model can be in one of
two situations: In the {\it degenerate case}
$\GGs^{\beta,\pi}_\mmTh=\{\Measu\}$ for all
$\beta\in\bR$ (recall that $\Measu:=\int \Measx\,\pi(d\ox)$),
and
consequently $\wp^{\beta,\pi}=-\beta\Measu(\Fio)$
for all $\beta$ and $I_\Fio(e)=0$ for $e=\Measu(\Fio)$
and $\infty$ otherwise.
Henceforth assume we are not in the
degenerate case. Then
the classes $\GGs^{\beta,\pi}_\mmTh$
of Gibbs measures are disjoint for all
distinct $\beta$, and in particular $\Measu$ is
the Gibbs measure only for $\beta=0$. To each
$\beta$ corresponds a unique nonempty closed interval
$[e_0(\beta),e_1(\beta)]$
of compatible energy values that can be characterized
in the following ways:
$$\aligned
[e_0(\beta),e_1(\beta)]&=\{e:
-\wp^{\beta,\pi}=\beta e+I_\Fio(e)\}\\
&=\{P(\Fio): P\in\GGs^{\beta,\pi}_\mmTh\}\\
&=\{e: \eta^c(e|\beta)=0\}.
\endaligned
\tag 9.12
$$
In the last formula, the rate function $\eta^c(e|\beta)$
is the one associated to $\mmg=\Phi$. Furthermore,
the intervals $[e_0(\beta),e_1(\beta)]$ are disjoint
for distinct $\beta$. (But $e_0(\beta)=e_1(\beta)$
for all but countably many $\beta$.)
Turning to (9.11), $I_\Fio$ is obviously convex and the set
$\{I_\Fio<\infty\}$ is a nondegenerate
bounded interval
that may or may not contain either of its endpoints.
If $e$ is an endpoint of $\{I_\Fio<\infty\}$, then
either $I_\Fio(e)=\infty$, or
$I_\Fio(e)<\infty$ and $I_\Fio$ has infinite
slope (negative for left, positive for right
endpoint) at $e$. To each interior point $e$ of
$\{I_\Fio<\infty\}$
corresponds a unique
$\beta=\beta(e)$ such that
$e_0(\beta)\leqq e\leqq e_1(\beta)$.
The infimum in (9.11) is
attained at this $\beta$, and $-\beta$ is the slope of
the unique tangent to
$I$ at $e$. The infimum in
(9.5) for $v=e$ and $g=\Fio$ is attained at
$Q$ if and only if $Q\in\GGs^{\beta(e),\pi}_\mmTh$.
The association $e\mapsto\beta(e)$ is
decreasing because $I_\Fio$ is convex,
with $\beta=0$ corresponding to $e=\Measu(\Fio)$.
$I_\Fio$ is differentiable on the interior
$\{I_\Fio<\infty\}^\circ$, and
$$\aligned
\{I_\Fio<\infty\}^\circ&=\bigcup_{\beta\in\bR}
[e_0(\beta),e_1(\beta)]\\
&\subset
\bigl[\,\int (\text{$\Measx$-ess inf $\Fio$})\,
\pi(d\ox)\,,\,
\int (\text{$\Measx$-ess sup $\Fio$})\,
\pi(d\ox)\,\bigr].
\endaligned
\tag 9.13$$
Also for a general observable $\mmg$,
the $\beta\leftrightarrow e$ duality clarifies
the relationship of the canonical and microcanonical
limit points (i.e. zeroes of the rate functions):
For all $\beta$,
$$\{v:\eta^c(v|\beta)=0\}=
\bigcup_{e_0(\beta)\leqq e\leqq e_1(\beta)}
\{v:\eta^{mc}(v|e)=0\}. \tag 9.14
$$
This is the `equivalence of canonical
and microcanonical ensembles.'
In particular, if $e=e_0(\beta)=e_1(\beta)$
is the unique
minimizer in (9.10), then
$$\{v: \eta^{mc}(v|e)=0\}=\{v: \eta^c(v|\beta)=0\},$$
and $|V_n|^{-1}g_n$ converges towards this set
exponentially fast under both microcanonical and
canonical probabilities.
Before concluding this section
with the proofs, let us note that all of the above
took place in the context of the
skew model. Recall that the quantity
$u^{\pi}(\mu)$ (see
Proposition 5.4) was defined
as a candidate for specific energy in the sample model,
its virtue being that it gave the basic variational
principle. But it is not clear whether this
function would be
involved in a meaningful conditioning of
the sample model on an energy surface.
\demo{Proof of Theorem 9.8} By (9.2) and the argument of
\cite{Or1, Proposition 3.1}, it suffices to prove
the theorem for the distributions of $\empRn(g)$ instead of
$g_n/|V_n|$. Then
the {\ldp}s of $(i)$ and $(iii)$
follow from (3.e) and (7.9), respectively, by the
contraction technique of large deviation theory
\cite{DS, Lemma 2.1.4}. Now consider $(ii)$.
(9.4) for large $n$ follows from the assumption
$I_\Fio(e)<\infty$ and part $(i)$ applied to $\mmPhi$.
For such $n$
$$\aligned
&\nVl\mmLa^{e,\ox}_{n,\delta}\lbrak g_n/|V_n|\in A\rbrak\\
=&\nVl\Measx\lbrak(g_n,H^0_n)/|V_n|\in A\times [e-\delta,e+\delta]\rbrak\\
&\quad -\nVl\Measx\lbrak H^0_n/|V_n|\in [e-\delta,e+\delta]\rbrak.
\endaligned\tag 9.15
$$
Let $\e>0$. By $(i)$ and $I_\Fio$'s lower
semicontinuity there exist $\delta_\e$
and $n_{\delta,\e}$ such that, for $\delta<\delta_\e$ and
$n>n_{\delta,\e}$, the last term in (9.15)
is within $\e$
of $-I_\Fio(e)$. Apply part $(i)$
to the observable $(\mmg,\mmPhi)$ to let $n\to\infty$
in (9.15). Then let $\delta\to 0$ and use the fact that
$$\lim_{\delta\to 0}\inf\Sb v\in \Abar \\ |x-e|\leqq\delta
\endSb
I_{(g,\Fio)}(v,x)=\inf_{v\in \Abar}I_{(g,\Fio)}(v,e),$$
a consequence of the lower semicontinuity and
compact sublevel sets of $I_{(g,\Fio)}$.
\qed
\enddemo
\demo{Proof of (9.10) and (9.11)} By
(5.10) and (9.5) for $\mmg=\mmPhi$,
$$-\wp^{\beta,\pi}=
\inf_{e\in\bR}\inf_{Q(\Fio)=e}
\{\beta\,Q(\Fio)+\mms^\pi(Q)\}
=\inf_{e\in\bR}
\{\beta e+I_\Fio(e)\}.$$
This is (9.10). (9.11) follows by convex duality.
\qed
\enddemo
\proclaim{9.16. Lemma}
If $\Measu\in\GGsbpiTh$ for some $\beta\ne 0$,
then $\GGsbpiTh=\{\Measu\}$ for all real $\beta$.
\endproclaim
\demo{Proof} By assumption,
$$\int g(\ox)\,\measx(f)\,\pi(d\ox)=\int g(\ox)\,G^{\beta,\os,\ox}_\La
(f)\,\Measu(d\os,d\ox)$$
for all $g\in\cbx$ and $f\in\CC_\La$,
so we deduce that
$$H_\La(\osLa,\ostilde_{\La^c}, \ox)=-{\beta}^{-1}\log
Z^{\beta,\ostilde,\ox}_\La$$
$\Measu$-a.s., for all $\La\sbsb\bzd$. It follows easily
that $\Measu\in\GGsbpiTh$ for all $\beta$.
Suppose $P\in\GGsbpiTh$ for some $\beta\ne 0$, and
$P\ne\Measu$ so that $\mms^\pi(P)>0$.
($\GGs^{0,\pi}_\mmTh=\{\Measu\}$ is immediate.) Then by
the variational principle
$$\beta P(\Fio)+\mms^\pi(P)=\beta\Measu(\Fio).$$
Pick $\beta'$ so that $(\beta-\beta')/\beta<0$.
Multiply above by $\beta'/\beta$ to get
$$\beta'P(\Fio)+\mms^\pi(P)=\beta'\Measu(\Fio)+
\mms^\pi(P)\cdot
({\beta-\beta'})/{\beta}<\beta'\Measu(\Fio).$$
This contradicts the variational
principle because $\Measu\in\GGs^{\beta',\pi}_\mmTh
$.
\qed
\enddemo
\proclaim{9.17. Lemma} Either the classes $\GGsbpiTh$
are disjoint for distinct real $\beta$,
or $\GGsbpiTh=\{\Measu\}$
for all real $\beta$ (the degenerate case).
\endproclaim
\demo{Proof}
In view of Lemma 9.16, it suffices to show that
$Q\in\GGsbpiTh\cap\GGs^{\beta',\pi}_\mmTh$ for $\beta\ne\beta'$
forces
$Q=\Measu$. As in the proof of Lemma 9.16, we deduce that
$$H_\La(\osLa,\ostilde_{\La^c}, \ox)=-(\beta'-\beta)^{-1}
\log
{Z^{\beta,\ostilde,\ox}_\La}/{Z^{\beta',\ostilde,\ox}_\La}$$
for $\measLax$-a.a. $\osLa$, $Q$-a.a.
$(\ostilde_{\La^c},\ox)$. From this $Q=\Measu$ follows readily.
\qed
\enddemo
\demo{9.18. Remark} The degenerate case can take
place even if $\Fio$ is not a.s. constant,
with or without quenched disorder.
Suppose $\SS=\{0,1\}$, and define
a function $f$ on $\SS^2$ by $f(1,0)=1$,
$f(0,1)=-1$, $f(0,0)=f(1,1)=0$. Then for any
$\os\in\OO$ and any $n$, $\sum_1^nf(\os_i,\os_{i+1})
\in\{-1,0,1\}$. For any measure $\mu$, the
only possible rate function for the laws
$\mu\{\frac 1n \sum_1^nf\circ\theta_i\in\cdot\}$
is the trivial one: $I(0)=0$ and $I(t)=\infty$
for nonzero $t$. In particular, we have the
degenerate case for the Gibbs measures of the
interaction
$$\Fi_A(\os)=\cases f(\os_i,\os_{i+1})
&\text{if $A=\{i,i+1\}$ for some $i$},\\
0 &\text{otherwise.}
\endcases
$$
\enddemo
We return to the
proofs of the section:
That $\wp^{\beta,\pi}=-\beta\Measu(\Fio)$ in the
degenerate case is immediate from the variational
principle of Theorem 8.1.
(9.11) implies then that
$I_\Fio(e)=0$ for $e=\Measu(\Fio)$
and $\infty$ otherwise.
\demo{Proof of (9.12)} $P\mapsto P(\Fio)$ maps
the
compact convex set $\GGsbpiTh$ into $\bR$, continuously
and affinely. Its image is some compact interval. Define
$e_0(\beta)$ and $e_1(\beta)$ so that
$[e_0(\beta),e_1(\beta)]=\{P(\Fio): P\in\GGsbpiTh\}$.
By Theorem 8.1,
$I_\Fio(P(\Fio))=\mms^\pi(P)$ for every Gibbs measure
$P$, hence
$-\wp^{\beta,\pi}=\beta e+I_\Fio(e)$
for all $e\in [e_0(\beta),e_1(\beta)]$. Conversely,
if $-\wp^{\beta,\pi}=\beta e+I_\Fio(e)$,
find $Q$ (using (9.5) and
the lower semicontinuity and compact
sublevel sets of $\mms^\pi$) such that $I_\Fio(e)=
\mms^\pi(Q)$ and $Q(\Fio)=e$. Then
$-\wp^{\beta,\pi}=\beta Q(\Fio) +\mms^\pi(Q)$,
and Theorem 8.1 implies that $Q\in\GGsbpiTh$.
We have proved the first two equalities of (9.12).
The last one is obvious from the definition (9.7).
\qed
\enddemo
To show that the sets $[e_0(\beta),e_1(\beta)]$
are disjoint for distinct $\beta$, suppose we have
$\beta\ne\beta'$,
$P\in\GGsbpiTh$, and $P'\in\GGs^{\beta',\pi}_\mmTh$
such that $P(\Fio)=P'(\Fio)$. By the variational
principle,
$$\beta P(\Fio)+\mms^\pi(P)\leqq
\beta P'(\Fio)+\mms^\pi(P'),$$
hence $\mms^\pi(P)\leqq \mms^\pi(P')$, and by
symmetry $\mms^\pi(P)=\mms^\pi(P')$. But then
Theorem 8.1 again implies $P'\in\GGsbpiTh$,
and by Lemma 9.17 we must be in the degenerate case,
which by assumption we are not.
The set $\{I_\Fio<\infty\}$ is an interval
by the convexity of $I_\Fio$, but not
a singleton because we have excluded the degenerate
case. For all $\beta$,
$$-\int \text{$\Measx$-ess sup $(\beta\Fio)$}\,
\pi(d\ox)\,\leqq \wp^{\beta,\pi}
\leqq \,-\int \text{$\Measx$-ess inf $(\beta\Fio)$}\,
\pi(d\ox),
$$
hence by (9.11)
$$\{I_\Fio<\infty\}\subset
\bigl[\,\int (\text{$\Measx$-ess inf $\Fio$})\,
\pi(d\ox)\,,\,
\int (\text{$\Measx$-ess sup $\Fio$})\,
\pi(d\ox)\,\bigr].
$$
Suppose $e$ is an interior point of
$\{I_\Fio<\infty\}$.
By convexity
$I_\Fio$ has a tangent at $e$, with slope $-\beta$
for some $\beta$. Then (9.11) forces
$-I_\Fio(e)=\beta e+\wp^{\beta,\pi}$, so in particular
$e\in [e_0(\beta),e_1(\beta)]$.
If the infimum in (9.11) were achieved at distinct
$\beta$ and $\beta'$, then $e$ would be an energy
value compatible with both $\beta$ and $\beta'$,
contradicting what was proved earlier.
That the infimum in (9.5) for $g=\Fio$ is
attained precisely on the set
$\{Q\in\GGs^{\beta(e),\pi}_\mmTh : Q(\Fio)=e\}$
is another consequence of the
variational principle.
Suppose $e$
is an endpoint of $\{I_\Fio<\infty\}$ with
$I_\Fio(e)<\infty$. If
$I_\Fio$ had finite slope at $e$, then it
would have infinitely many tangents at $e$
(because $I\equiv\infty$ on one side of $e$),
again associating $e$ to multiple $\beta$,
in contradiction to previous conclusions.
The uniqueness of the tangent also
implies that $I_\Fio$ is differentiable
on $\{I_\Fio<\infty\}^\circ$.
(9.13)
is evident by now.
\demo{Proof of (9.14)} Suppose $\eta^c(v|\beta)=0$.
Let $Q$ be a minimizer in (9.7)
and $e:=Q(\Fio)$.
Then
$$0=\beta e+\inf\Sb Q(g)=v\\ Q(\Fio)=e\endSb\mms^\pi(Q)
+\wp^{\beta,\pi},$$
so an application of (9.5) to the observable
$(\mmg,\Phi)$ gives
$$-\wp^{\beta,\pi}=\beta e+I_{(g,\Fio)}(v,e).\tag 9.19$$
(9.10), (9.19), and $I_\Fio(e)\leqq
I_{(g,\Fio)}(v,e)$
combine to give
$-\wp^{\beta,\pi}=\beta e+I_\Fio(e)$ and
$I_{(g,\Fio)}(v,e)=I_\Fio(e)$, so that
$e\in [e_0(\beta),e_1(\beta)]$ and
$\eta^{mc}(v|e)=0$.
Conversely, assume $e\in [e_0(\beta),e_1(\beta)]$
and $\eta^{mc}(v|e)=0$.
These imply (9.19). Using (9.5), pick a $Q$ such that
$Q(g,\Fio)=(v,e)$ and $I_{(g,\Fio)}
(v,e)=\mms^\pi(Q)$.
Substitute these into (9.19) and then into
(9.7) to get
$\eta^c(v|\beta)=0$.
\qed
\enddemo
\vskip .4in
%\input sec10
\flushpar
{\bf 10. \ Equivalence of Ensembles for Measures}
\hbox{}
\flushpar
In section 9 we deduced equivalence of the microcanonical
and canonical ensembles by looking at the possible
limit points of an observable in the infinite volume
limit. In this final section of the paper we
do the same for measures.
We shall show
that, in an appropriate sense,
the limit points of the microcanonical
probability measures are Gibbs measures at the
temperature associated with the energy of the
microcanonical ensemble. Our approach to
this question via large deviation theory is
motivated by section 3 of \cite{DSZ},
where this program was carried out
for invariant interactions.
For the duration of the section, fix $\pi\in\mtx$ and
a real number $e$ such that $I_\Fio(e)<\infty$.
To avoid worrying about whether $e$ is an interior
point or a
boundary point of $\{I_\Fio<\infty\}$,
adopt the following convention: If $e$ is the left
endpoint of $\{I_\Fio<\infty\}$ and $I_\Fio(e)<\infty$,
then set $\beta(e)=\infty$ and
$$\GGs^{\infty,\pi}_\mmTh=\{Q\in\MSOOXX:
Q(\Fio)=e\,,\, \mms^\pi(Q)=I_\Fio(e)\,\}.$$
Similarly for a right endpoint. Then all
real numbers $e$ such
that $I_\Fio(e)<\infty$ have a uniquely
defined inverse temperature $-\infty\leqq\beta(e)
\leqq\infty$ and there is a well-defined nonempty
class of Gibbs measures at
inverse temperature $\beta(e)$.
Set
$$\KKs:=\{Q\in \GGs^{\beta(e),\pi}_\mmTh:
Q(\Fio)=e\}.$$
We know from section 9 that $\KKs$ is the nonempty
compact convex set of measures $Q$ for which
$Q(\Fio)=e$ and $\mms^\pi(Q)=I_\Fio(e)$.
Since the microcanonical measures are doubly
indexed by $\delta>0$ and $n\in\bN$,
we must be precise
about passing to limits. Let us say that a
class $\{\mu_{n,\delta}\}$ of measures is
{\it relatively compact as first $n\to\infty$
and then $\delta\to 0$ } if
\roster
\item"{(i)}" $\{\mu_{n,\delta}\}_{n=1}^\infty$
is relatively compact for each fixed $\delta$,
\endroster
and
\roster
\item"{(ii)}" whenever $\delta_j\to 0$
and $\nu_{\delta_j}$ is a limit point
of $\{\mu_{n,\delta_j}\}_{n=1}^\infty$ for each
$j$, the
sequence
$\{\nu_{\delta_j}\}_{j=1}^\infty$
is relatively compact.
\endroster
Say {\it $\nu$ is a limit point of $\{\mu_{n,\delta}\}$}
(as first $n\to\infty$
and then $\delta\to 0$)
if some such sequence $\{\nu_{\delta_j}\}_{j=1}^\infty$
converges to $\nu$.
We shall first consider
averaged microcanonical measures
$$\mmGaexnd:=\nVsum \mmLaexnd\circ\theta_{-\ii}.$$
Define a function $\mmk^e$ from $\MOOXX$ into
$[0,\infty]$ by
$$\mmk^e(Q):=\cases \mms^\pi(Q)-I_\Fio(e)
&\text{if $Q$ is invariant and $Q(\Fio)=e$},\\
\infty &\text{otherwise.}
\endcases
$$
\proclaim{10.1. Theorem} Suppose $\ox$ is
generic for $\pi$.
\roster
\item"{$(i)$}" As first $n\to\infty$ and then
$\delta\to 0$, the distributions
$\mmLaexnd\{\empRn\in\ett\}$ satisfy a {\ldp}\
with rate
function $\mmk^e$, as in {\rm (9.9)}.
\pikkuhyppy
\item"{$(ii)$}" The probability measures
$\mmGaexnd$ are relatively compact as first
$n\to\infty$ and then
$\delta\to 0$. If $\Ga$ is a limit point, then
there is a probability measure $\a$ on $\KKs$
such that
$$\Ga=\int_{\KKs} Q\,\a(dQ)$$
and $\a$ is a limit point of the laws
$\mmLaexnd\{\empRn\in\ett\}$.
In particular,
$\Ga$ is an element of $\KKs$.
\endroster
\endproclaim
Our techniques do not permit us to tackle
individual microcanonical measures $ \mmLaexnd$.
We can replicate Theorem 10.1
for the $\pi$-a.s. defined map $\Laepnd:\ox\mapsto
\Laexnd$, where $\Laexnd$ is the $\OO$-marginal
of $\mmLaexnd$. But to do so requires some
technical assumptions and changes:
Assume now that $\XX=E^{\bzd}$ for some Polish space
$E$, $\bzd$ acts on $\XX$ by translations, and
$\measox=\measo{\ox_\bold0}$ depends only on the
$\ox_\bold0$-coordinate.
Furthermore, assume $\pi$ is an i.i.d. measure
on $\XX$. Define the stationary empirical field
$\empRs$ by
$$\empRs:= \nVsum \delta_{\mmtheta_{\ii}
(\os^{(n)},\ox^{(n)})},$$
where $\os^{(n)}$ and $\ox^{(n)}$ are the periodized
configurations defined as in the proof of Theorem 7.9.
Redefine the microcanonical probability measure
by
$$\mmLa^{e,\ox}_{n,\delta}(A):=
\Measx\lbrak A\,\big|\;
|\,\empRs(\Fio)-e\,|\leqq\delta \rbrak.
$$
The advantage of this definition is that the
conditioning event now depends only on
$(\os_\ii,\ox_\ii: \ii\in V_n)$.
Consider $\Laepnd$ as an element of the space
$\MM^{\pi,\piste}$ of $\pi$-a.s. defined measurable
maps from $\XX$ into $\MOO$, introduced in section 6.
Give this space a Polish topology by identifying
a map $\varrho^\piste$ with the measure
$\varrho^\ox(d\os)\,\pi(d\ox)$ on $\OOXX$. In other
words, $\varrho_n^\piste\to\varrho^\piste$
in $\MM^{\pi,\piste}$ as
$n\to\infty$ if
$$\lim_{n\to\infty}
\int \varrho_n^\ox(f^\ox)\,\pi(d\ox)=
\int \varrho^\ox(f^\ox)\,\pi(d\ox)
$$
for all bounded continuous functions $f$ on $\OOXX$,
where $f^\ox(\os):= f(\os,\ox)$. Set
$$\aligned
\KK:=&\lbrak
\gamma^\piste
\in \GGpipTh:\,
\int \gamma^\ox(\Fio^\ox)\,\pi(d\ox) =e\rbrak\\
=&\lbrak \varrho^\piste\in \MM^{\pi,\piste}:
\text{ $\int \varrho^\ox(\Fio^\ox)\,\pi(d\ox) =e$
and $h(\varrho^\piste|\lambda^\piste)=I_\Fio(e)$ }
\rbrak.
\endaligned
$$
$h(\varrho^\piste|\lambda^\piste)$ is
the entropy defined in (8.6). Recall the
bijection $Q\mapsto q^\piste$ from $\MSpiOOXX$
onto $\MMpipTh$ of Lemma 6.3$(ii)$.
It restricts to a homeomorphism from $\KKs$ onto $\KK$.
With these assumptions
we can state our final theorem:
\proclaim{10.2. Theorem}
The maps $\Laepnd$ are relatively compact as first
$n\to\infty$ and then
$\delta\to 0$. If $\varrho^\piste$
is a limit point, then
there is a probability measure $\a$ on $\KKs$
such that
$$\varrho^\piste=\int_{\KKs} q^\piste
\,\a(dQ)$$
and $\a$ is a limit point of the laws
$\int \mmLaexnd\{\empRn\in\ett\}\,\pi(d\ox)$.
In particular,
$\varrho$ is an element of $\KK$.
\endproclaim
We now turn to the proofs.
For $\delta\geqq 0$
and $Q\in\MOOXX$, set
$$m(\delta):=\inf_{|Q(\Fio)-e|\leqq\delta}
\mms^\pi(Q),$$
$$\KKs_\delta:=\{Q\in\MOOXX:
\mms^\pi(Q)=m(\delta)\,,\, |Q(\Fio)-e|\leqq\delta\},
$$
and
$$\mmk^e_\delta(Q):=\cases \mms^\pi(Q)-m(\delta)
&\text{if $Q$ is invariant and
$|Q(\Fio)-e|\leqq\delta$},\\
\infty &\text{otherwise.}
\endcases
$$
In this new notation, $m(0)=I_\Fio(e)$,
$\KKs_0=\KKs$, and $\mmk^e_0=\mmk^e$.
\proclaim{10.3. Lemma}
\roster
\item"$(i)$" $m(\delta)$ is a continuous,
convex,
decreasing function from $[0,\infty)$ onto
$[0,I_\Fio(e)]$.
\pikkuhyppy
\item"$(ii)$" The sets $\KKs_\delta$ are nonempty, convex, and
compact, and if $G$ is any open
neighborhood of $\KKs_\delta$,
then $\KKs_{\delta'}\subset G$ for $\delta'$ close
enough to $\delta$. Moreover, the union
$\KKs_{0,\infty}:=\bigcup_{\delta\geqq 0}\KKs_\delta$
is compact.
\pikkuhyppy
\item"$(iii)$" Fix $\delta>0$ and suppose $\ox$
is generic for $\pi$.
Then the distributions
$\mmLaexnd\{\empRn\in\ett\}$
satisfy a \ldp\ with rate $\mmk^e_\delta$,
as $n\to\infty$.
\pikkuhyppy
\item"$(iv)$" Suppose $\ox$
is generic for $\pi$. The distributions
$\mmLaexnd\{\empRn\in\ett\}$
are relatively compact as first $n\to\infty$
and then $\delta\to 0$,
and all the limit points are supported
on $\KKs$.
\endroster
\endproclaim
\demo{Proof} $(i)$ $m(\delta)$ is obviously
convex and decreasing,
and $m(\delta)=0$ for large $\delta$.
To prove continuity from the right: Let $\delta_n
\searrow\delta$. Pick $Q_n$'s such
that $|Q_n(\Fio)-e|\leqq\delta_n$
and $\mms^\pi(Q_n)=m(\delta_n)$,
using the compact
sublevel sets and lower semicontinuity of $\mms^\pi$. The $Q_n$'s
lie inside the compact set $\{ \mms^\pi\leqq m(0)\}$,
hence we may pass to a convergent subsequence and assume
$Q_n\to Q$. Then $|Q(\Fio)-e|\leqq\delta$,
and by lower semicontinuity
$$m(\delta)\leqq \mms^\pi(Q)\leqq \liminf_{n\to\infty}
\mms^\pi(Q_n)
=\liminf_{n\to\infty} m(\delta_n).$$
This suffices for continuity from the right, for
$\DDD{m(\delta)\geqq \limsup_{n\to\infty} m(\delta_n)}$
by the monotonicity.
Now suppose $\delta_n\nearrow\delta$. Pick $Q$
and $Q_1$ that realize the
infima in the definitions of $m(\delta)$ and $m(\delta_1)$,
respectively.
Let $t_n\nearrow 1$ be such that $\delta_n=t_n\delta+(1-t_n)
\delta_1$. By the convexity of $\mms^\pi$,
$$\limsup_{n\to\infty} m(\delta_n)\leqq \limsup_{n\to\infty}
\lbrak t_n\mms^\pi(Q)+(1-t_n)
\mms^\pi(Q_1)\rbrak =m(\delta).$$
By monotonicity again we have continuity from the left.
$(ii)$ The first clause is true by the properties
of $\mms^\pi$ from (3.e).
Suppose $\delta_n\to\delta$ and $Q_n\in\KKs_{\delta_n}
\erot G$ for some open neighborhood $G$ of $\KKs_\delta$.
As above, we may assume $Q_n\to Q$.
Then $Q\in G^c$. But $\mms^\pi(Q_n)=m(\delta_n)$ for all
$n$, so by $\mms^\pi$'s lower semicontinuity and part $(i)$,
$\mms^\pi(Q)\leqq m(\delta)$. And
$|Q(\Fio)-e|\leqq\delta$ by
the continuity of the integral $Q(\Fio)$, hence we must
have $\mms^\pi(Q)=m(\delta)$ and $Q\in\KKs_\delta$,
a contradiction with $Q\in G^c$.
$\KKs_{0,\infty}$ is a subset
of the compact set $\{\mms^\pi\leqq m(0)\}$,
so we need only prove closedness. Suppose
$Q_n\in\KKs_{\delta_n}$ and $Q_n\to Q$. If the
$\delta_n$ remain bounded, pass to a convergent
subsequence $\delta_{n'}\to\delta$, and then
$Q\in\KKs_{\delta}$ by the previous paragraph.
If $\delta_{n'}\nearrow\infty$ for some subsequence
$n'$, then $Q_{n'}=\Measu$ for large enough $n'$
and $Q=\Measu\in\KKs_{0,\infty}$.
For $(iii)$, let $0<\eta<\delta$ and write for large $n$
$$\aligned
&\nVl\Measx\{\empRn\in A^\circ , |\empRn(\Fio)-e|<\delta-\eta\}
-\nVl \Measx\{|\empRn(\Fio)-e|\leqq\delta+\eta\}\\
\leqq\, &\nVl\mmLaexnd\{\empRn\in A\}\\
\leqq\, &\nVl\Measx\{\empRn\in \Abar , |\empRn(\Fio)-e|\leqq\delta+\eta\}
-\nVl \Measx\{|\empRn(\Fio)-e|<\delta-\eta\}.
\endaligned
$$
For the legitimacy of this
deduce from (3.e) that
$$\lil \Measx\{|\empRn(\Fio)-e|<\delta-\eta\}\geqq
-m(\delta-\eta/2)>-\infty.$$
Let $n\to\infty$ and then $\eta\searrow 0$,
and use the \ldp\ of (3.e), the
lower semicontinuity and compact sublevel sets of $\mms^\pi$,
and the continuity of $m(\delta)$. The upper bound
follows without difficulty, but the lower bound
emerges in the form
$$\lil\mmLaexnd\{\empRn\in A\}\geqq
-\lim_{\eta\searrow 0}\;
\inf
\Sb Q\in A^\circ \\ |Q(\Fio)-e|<\delta-\eta \endSb
\mms^\pi(Q)+m(\delta).
$$
Thus we need to show that
$$\lim_{\eta\searrow 0}\;\inf
\Sb Q\in A^\circ \\ |Q(\Fio)-e|<\delta \endSb
\mms^\pi(Q)\leqq
\inf
\Sb Q\in A^\circ \\ |Q(\Fio)-e|\leqq\delta \endSb
\mms^\pi(Q).
$$
Let $Q'\in A^\circ$ be such that
$|Q'(\Fio)-e|\leqq\delta$ and $\mms^\pi(Q')<\infty$.
(If no such $Q'$ exists, we can stop here.) Pick
$P$ such that $|P(\Fio)-e|\leqq\delta/2$ and
$\mms^\pi(P)<\infty$. Such a $P$ exists because
$m(\delta/2)\leqq m(0)<\infty$ by assumption.
Pick $00$,
are supported by $\KKs_\delta$
(see the remark on p. 49 of
\cite{Pa}). All the limit points for all $\delta>0$
are supported by the compact set $\KKs_{0,\infty}$,
hence also part (ii) of the definition
is satisfied. The last statement of $(iv)$ follows from
part $(ii)$ since the sets $\KKs_\delta$
converge to $\KKs$ as $\delta\searrow 0$.
\qed
\enddemo
\demo{Proof of Theorem 10.1} Part $(i)$ follows from
Lemma 10.3 by letting $\delta\searrow 0$ in part $(iii)$.
Part $(ii)$ follows from Lemma 10.3$(iv)$ because
$\mmGaexnd=\mmLaexnd(\empRn(\ett))$ is the image of
$\mmLaexnd\{\empRn\in\ett\}$ under the continuous map that
sends a distribution on probability measures to its
expectation.
\qed
\enddemo
\demo{Proof of Theorem 10.2}
Suppose $f$ is a bounded measurable function
on $\OOXX$ that depends only
on $(\os_\ii,\ox_\ii: \ii\in V_n)$.
Then
$$\aligned
\int \Laexnd(f^\ox)\,\pi(d\ox)
&= \int \mmLa^{e,\ox^{(n)}}_{n,\delta}
(f)\,\pi(d\ox)
= \int \mmLa^{e,\ox^{(n)}}_{n,\delta}
(\empRs(f))\,\pi(d\ox)\\
&= \int \mmLaexnd(\empRs(f))\,\pi(d\ox).
\endaligned
\tag 10.4
$$
Theorem 10.2 now follows as Theorem 10.1 did because
the distributions
$\mmLaexnd\{\empRs\in\ett\}$ and
$\mmLaexnd\{\empRn\in\ett\}$ satisfy the
same {\ldp}s. In particular, the
distributions $\mmLaexnd\{\empRs\in\ett\}$
concentrate on $\KKs$
$\pi$-a.s. as first $n\to\infty$
and then $\delta\searrow 0$, hence so do the distributions
$\int \mmLaexnd\{\empRs\in\ett\}\,\pi(d\ox)$.
And by (10.4), this determines the asymptotic
behavior of $\Laepnd$ in the topology
of $\MM^{\pi,\piste}$.
\qed
\enddemo
\vskip .4in
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\enddocument