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\documentstyle[11pt]{article}
\author{Hans-Otto Georgii \\
{\small Mathematisches Institut der Universit\"at M\"unchen}\\
{\small Theresienstr. 39, D--80333 M\"unchen}}
\title{Large deviations and the equivalence of ensembles \\for Gibbsian particle
systems with superstable interaction}
\date{}
\textwidth 14.5cm \textheight 22cm
\oddsidemargin 7mm \evensidemargin -1mm \topmargin 1mm
\parindent 0.5cm
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\sloppy
\def\R{{\bf R}}
\def\Z{{\bf Z}}
\def\skip{\medskip\smallskip}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray*}}
\def\eea{\end{eqnarray*}}
\def\proof{{\em Proof.} \enspace}
\def\ti{\to\infty}
\def\eps{\varepsilon}
\def\eh{\mbox{\small $\frac{1}{2}$}}
\def\Pinv{{\cal P}_{\Theta}}
\def\Pinvquadr{{\cal P}_{\Theta}^{(2)}}
\def\pinp{P \in {\cal P}_{\Theta}}
\def\ominOm{\omega \in \Omega}
\def\zbet{z,\beta}
\def\nzbet{n,z,\beta}
\def\free{\mbox{\scriptsize free}}
\def\per{\mbox{\scriptsize per}}
\def\bc{\mbox{\scriptsize bc}}
\def\nfree{n,\free}
\def\nper{n,\per}
\def\nbc{n,\bc}
\def\vni{v_n^{-1}}
\def\xinom{x \in \omega}
\def\Lan{\Lambda_n}
\def\uLan{\cap \Lan}
\def\thx{\vartheta_x}
\def\vFR{v_n F(R_n)}
\def\lslog{\limsup_{n \to \infty} \vni \log \,}
\def\lilog{\liminf_{n \to \infty} \vni \log \,}
\def\lsc{\mbox{\scriptsize lsc}}
\def\usc{\mbox{\scriptsize usc}}
\def\taul{\tau_{\cal L}}
\def\rhoquadr{\rho^{(2)}}
\def\micro{Q_{n|D,\eps,\per}}
\def\grand{P_{\nzbet,\per}}
\begin{document}
\maketitle
\noindent
{\bf Summary.}\enspace For Gibbsian systems of particles in $\R^d$, we
investigate large deviations of the stationary empirical fields in increasing
boxes. The particle interaction is given by a superstable, regular pair potential.
The large deviation principle is established for systems with free or periodic
boundary conditions and, under a stronger stability hypothesis on the potential,
for systems with tempered boundary conditions, and for tempered (infinite-volume)
Gibbs measures. As a by-product we obtain the Gibbs variational formula for the
pressure. We also prove the asymptotic equivalence of microcanonical and
grand canonical Gibbs distributions and establish a variational expression
for the thermodynamic entropy density.
\skip
\skip\noindent
{\em Mathematics subject classification (1991).} 60F10, 60G55, 60K35, 82B05,
82B21.
\skip
\skip
\section{Introduction}
One of the classical themes of Equilibrium Statistical Mechanics is the study
of the fluctuations of extensive quantities, such as the particle numbers and
the energies of particle configurations in finite boxes, in the infinite-volume
limit. This includes the problems of existence and variational characterization
of the pressure, and of the asymptotic equivalence of the Gibbs ensembles (on
the level of measures). Whereas these questions are fairly well understood in
the case of lattice systems (see \cite{G} and the references therein), the
situation is less satisfactory in the case of continuous systems of particles
in Euclidean space. In this paper we develop a large deviation theory for such
particle systems, with the aim of contributing to a systematic study of the
questions above.
The general setting is as follows.
We consider the Euclidean space $\R^d$ of any dimension $d \geq 1$. A configuration
of particles (without multiple occupancies) is described by a locally finite
subset $\omega$ of $\R^d$, i.e., a set $\omega \subset \R^d$ having
finite intersection with every bounded set. We write $\Omega$ for the set of all
such configurations $\omega$. $\Omega$ is equipped with the $\sigma$-algebra
$ {\cal F}$ generated by the counting variables $N_B: \omega \to {\rm card}(\omega \cap
B)$ for Borel subsets $B$ of $\R^d$. It is well-known \cite{GZ,KMM} that
${\cal F}$ is the Borel ${\sigma}$-algebra for a natural Polish toplogy on
$\Omega$. The translation group $\Theta = (\thx)_{x \in \R^d}$ acting
on $(\Omega, {\cal F})$ is defined by $\thx \omega = \{ y-x:y \in \omega \},
\, \ominOm, \, x \in \R^d$. The mapping $(x,\omega) \to \thx \omega$ is known
to be measurable \cite{KMM}. We let ${\cal P}$ denote the set of all probability measures
$P$ on $(\Omega,{\cal F})$ with finite expected particle numbers
$P(N_{\Delta})\equiv\int N_{\Delta} dP$ in bounded Borel sets $\Delta \subset
\R^d$, and we write $\Pinv$ for the set of all $\Theta$-invariant
$P \in {\cal P}$.
For each $\pinp$ there exists a number $\rho(P)<\infty$,
the {\em intensity} of $P$,
such that $P(N_{\Delta}) = \rho(P) \, |\Delta | $ for all Borel sets $\Delta$.
Here $|\Delta |$ is the Lebesgue measure of $\Delta$.
We introduce a topology $\taul$ on ${\cal P}$ as follows. Let ${\cal L}$ denote the
class of all measurable functions $f:\Omega \to \R$ which are {\em local},
in that $f(\omega) = f(\omega \cap \Delta)$ for some bounded Borel set $\Delta$
and all $\ominOm$, and {\em tame}, in that $|f| \leq c (1+N_{\Delta})$ for
(without loss the same) $\Delta$ and some constant $c<\infty$. The {\em topology}
$\taul$ {\em of local convergence} is then defined as the weak$^{\star}$
topology on ${\cal P}$ relative to ${\cal L}$, i.e., as the smallest topology
on ${\cal P}$ making the mappings $P \to P(f)\equiv \int f \, dP$ continuous. In
particular, the mappings $P \to P(N_{\Delta})$ for bounded Borel sets
$\Delta$ are continuous relative to $\taul$. This shows that $\taul$ is much
finer than the usual weak topology on ${\cal P}$ which is based on the
above-mentioned Polish toplogy on $\Omega$.
For each $n \geq 0$, we consider the half-open cube $\Lan = [-n-\eh,n+\eh [\,^d$
of volume $v_n = (2n+1)^d$ and the associated {\em invariant empirical field}
$$
R_{n,\omega} = \vni \int_{\Lan} \delta_{\thx \omega^{(n)}} dx \eqno(1.1)
$$
of any configuration $\ominOm$. In (1.1), we replaced $\omega$ by the
$\Lan$-periodic continuation
$$
\omega^{(n)} = \{ x+(2n+1)i:\xinom \uLan, \, i \in \Z^d \}
$$
of its restriction to $\Lan$. This has the advantage of making $R_{n,\omega}$
translation invariant. Thus $R_{n,\omega} \in \Pinv$ for all $n$ and $\omega$.
If $\Pinv$ is equipped with the evaluation ${\sigma}$-algebra generated by the
maps $P \to P(A), \, A \in {\cal F}$, the random measure $R_n: \omega \to
R_{n,\omega}$ becomes a measurable mapping from $\Omega$ to $\Pinv$. The asymptotic
behavior of the empirical fields can be described as follows: For each $\pinp$,
$$
\taul\mbox{ -}\lim_{n \to \infty} R_n = P^{\cdot}_{\cal I} \qquad \mbox{in }
P\mbox{-probability,} \eqno(1.2)
$$
where $\omega \to P^{\omega}_{\cal I}$ is a regular conditional probability of
$P$ relative to the $\sigma$-algebra ${\cal I}$ of ${\Theta}$-invariant sets in
${\cal F}$. This follows immediately from Wiener's multidimensional mean ergodic
theorem, cf. \cite{GZ}.
In this paper we study large deviations from the ergodic theorem (1.2) when
$P$ is \linebreak Gibbsian relative to a suitable pair interaction $\varphi$. More
precisely, we establish a large deviation principle for $R_n$ when the particles
are distributed according to a Gibbs distribution in $\Lan$ with free or
periodic boundary condition and the underlying potential $\varphi$ is superstable
and satisfies a decay condition called regularity (see Theorem 2). Under the stronger
hypothesis that $\varphi$ diverges at the origin sufficiently fast,
we obtain a uniform large deviation principle relative to
Gibbs distributions with boundary configurations $\zeta \in \Omega$
satisfying a uniform condition of temperedness (Theorem 3(a)).
By virtue of the well-known superstability estimates of Ruelle \cite{Rb}, this
leads to a large deviation principle for tempered Gibbs measures
on $(\Omega,{\cal F})$ (Theorem 3(b)). The rate function is, of course, given by the excess
of the free energy density over its equilibrium value; the latter is given by
the pressure. A basic ingredient of all this is the existence and lower
semicontinuity of the (internal) energy density (Theorem 1). Finally,
we prove
a limit theorem for conditional Poisson distributions of microcanonical type,
implying the equivalence of Gibbs ensembles on the level of measures (Theorem 4).
This is an instance of the maximum entropy principle and is closely related to
a (microcanonical) Gibbs variational formula for the thermodynamic entropy
density.
In the Poissonian case of no interaction, the analoguous results were obtained earlier
in \cite{GZ}. Our results here rely heavily on this paper. A weak version of
a large deviation principle for particles with superstable interaction of
finite range is also contained in \cite{OVY}. The case of particles with hard core
(which is contained in the present work) was already treated in \cite{Ghc}.
\section{Statement of results}
We begin describing the particle interactions which we will consider. We assume,
for simplicity, that the interaction is only pairwise and thus given by an even
measurable function $\varphi :\R^d \rightarrow \R \cup \{ \infty \}$.
Such a $\varphi$ is called a {\em potential\/}. For each $n \geq 0$,
$$
H_{n}(\omega)\equiv H_{n,\free}(\omega) = \eh
\sum_{x,y \in \omega \cap \Lambda_{n} , x \not= y} \varphi(y-x) \, ,\qquad
\quad \omega \in \Omega, \eqno(2.1)
$$
is called the associated {\em Hamiltonian } in $\Lambda_n$ with
{\em free boundary condition}.
A potential $\varphi$ is said to be {\em stable} if there exists a constant
$b = b(\varphi) < \infty$ such that
$$
H_n \geq - b N_n \qquad \mbox{for all} \enspace n \geq 0. \eqno(2.2)
$$
Here $N_n = N_{\Lambda_{n}}$. In particular, (2.2)
implies that $\varphi \geq -2 b$. Sufficient conditions for $\varphi$ to be
stable can be found in \cite{RBuch} . Let us say that $\varphi$ is
{\em purely repulsive} if $\varphi$ is nonnegative and bounded away from zero
near the origin, i.e., if there exists some $\delta = \delta(\varphi) > 0$
such that
$$
\varphi \geq \delta \, 1_{\{|\,\cdot\,| \le \delta \}}. \eqno(2.3)
$$
Here and below, $|\cdot|$ stands for the maximum norm on $\R^d$. A potential
$\varphi$ is called {\em superstable} if
$$
\varphi = \varphi^s + \varphi^r \eqno(2.4)
$$
for a stable $\varphi^s$ and a purely repulsive $\varphi^r$. The use of this
concept was revealed by the pioneering work of Ruelle \cite{Ra,Rb}.
Besides the hypothesis of superstability which assures that large particle
numbers in a bounded region require a large amount of energy we shall also
need a condition on the decay of $\varphi$. A potential $\varphi$ is called
{\em lower regular} if there exists a decreasing function
$\psi : [0,\infty[ \to [0,\infty[$ such that
$$
\varphi(x) \geq - \psi(|x|) \qquad \mbox{for all} \enspace
x \in \R^d \eqno(2.5)
$$
and
$$
\int_{0}^{\infty} \psi (s) s^{d-1} ds < \infty. \eqno(2.6)
$$
$\varphi$ will be called {\em regular} if $\varphi$ is lower regular and, in
addition, there exists some $r(\varphi) < \infty$ such that
$$
\varphi(x) \leq \psi(|x|) \qquad \mbox{whenever} \enspace
|x| \geq r(\varphi). \eqno(2.7)
$$
Our first result is the existence of the energy density of any $P \in
{\cal P}_{\Theta}$. To state it we need to recall that the {\em Palm measure}
of $\pinp$ is defined as the unique finite measure $P^{\circ}$
on $(\Omega,{\cal F})$
satisfying
$$
\int P(d\omega) \sum_{x \in \omega} f(x,\vartheta_x \omega) =
\int dx \int P^{\circ}(d \omega) \, f(x,\omega) \eqno(2.8)
$$
for all measurable functions $f:\R^d \times \Omega \to [0,\infty[$. We have
$P^{\circ}(\Omega) = P^{\circ}(\{\ominOm:\omega \ni 0 \})\linebreak = \rho(P)$,
and the normalized Palm measure $\rho(P)^{-1}\,P^{\circ}$ can be viewed
as the natural version of the conditional probability
$P(\,\cdot\, |\{\ominOm:\omega \ni 0\})$; see \cite{KMM} for more details. Let us
introduce the set
$$
\Pinvquadr = \lbrace \pinp: P(N^2_{\Delta}) < \infty \enspace
\mbox{for all bounded
Borel sets} \enspace \Delta \rbrace \eqno(2.9)
$$
of all second-order elements of $\Pinv$. For each $n \geq 0$ and $\pinp$ we let
$$
\Phi_n(P) = v_n^{-1}\,P(H_n), \eqno(2.10)
$$
be the expected {$\varphi$}-energy per volume in $\Lambda_n$. In view of (2.2),
$\Phi_n$ is well-defined (possibly equal to $+\infty$), and
$\Phi_n \geq -b \rho$. In Section 3 we shall prove the following.
\skip
\\
{\bf Theorem 1} \enspace {\em Suppose $\varphi$ is superstable and lower
regular. Then, for
each $\pinp$, the limit $\Phi(P) = \lim_{n \to \infty} \Phi_n(P)$ exists and satisfies}
$$
\Phi(P) = \left\{ \begin{array}{ll} P^{\circ}(f_{\varphi}) & \mbox{if}
\enspace P \in \Pinvquadr\\ \infty & \mbox{otherwise,} \end{array} \right.
\eqno(2.11)
$$
{\em where}
$$
f_{\varphi}(\omega) = \eh \sum_{0 \not= y \in \omega} \varphi(y) \, ,\qquad
\ominOm.\eqno(2.12)
$$
{\em Moreover, the function $\Phi : \Pinv \to \R \cup \{\infty\}$ is lower
semicontinuous (relative to $\tau_{\cal L}$).}
\skip
$\Phi(P)$ is called the {\em energy density} of $P$. It is clear that $\Phi$
is affine.
To state our large deviation results we next need to introduce the entropy
density. We let $Q \in \Pinv$ denote the Poisson point random field on
$\R^d$ with intensity $\rho(Q)=1$. For each $\pinp$ and $n \geq 0$
we write
$$
P_n = P(\{\ominOm: \omega \cap \Lambda_n \in \cdot \,\})
$$
for the restriction of $P$ to $\Lambda_n$. We think of $P_n$ as an element
of ${\cal P}$ which is supported on $\Omega_n = \{\ominOm: \omega \subset
\Lambda_n\}$. The negative {\em entropy density} of $P$ is then defined as the
(existing) limit
$$
I(P) = \lim_{n \to \infty} v_n^{-1} \, I(P_n;Q_n), \eqno(2.13)
$$
where $I(\,\cdot\, ;\,\cdot\,)$ stands for the relative entropy; see \cite{GZ} for
more details. $I$ is an affine function with $\tau_{\cal L}$-compact level
sets \cite{GZ}.
Suppose now we are given an inverse temperature $\beta > 0$ and an activity $z > 0$.
(Later on, we shall assume without loss that the units are chosen so that
$\beta =z =1$.) The {\em excess free energy density} of $\pinp$ is then given by
$$
I_{\zbet}(P) = I(P) + \beta \Phi(P) - \rho(P)\, \log z + p(\zbet), \eqno(2.14)
$$
where
$$
p(\zbet) = - \min \bigl [ I + \beta \Phi - \rho\, \log z \bigr ] .
\eqno(2.15)
$$
Theorem 2 below asserts that $p(\zbet)$ is nothing other than ($\beta$ times)
the pressure, and $I_{\zbet}$ is the rate function in a large deviation principle
for the distribution of the empirical fields $R_n$ under the Gibbs distributions
with free or periodic boundary conditions. (Configurational boundary conditions
will be considered later in Theorem 3.)
For $n \geq 0$, the {\em Hamiltonian} in $\Lambda_n$ with {\em periodic
boundary condition} is defined by
\setcounter{equation}{15}
\begin{eqnarray}
H_{n,\per}(\omega) & = & v_n \Phi(R_{n,\omega}) \nonumber \\
& = & \eh \sum_{x \in \omega \cap \Lambda_n,
y \in \omega^{(n)}, y \not= x}
\varphi(y-x).
\end{eqnarray}
The last equation follows from (2.11) and the easily verified fact that the
Palm measure of $R_{n,\omega}$ is given by
$$
R_{n,\omega}^{\circ} = \vni \sum_{\xinom \uLan} \delta_{\thx \omega^{(n)}},
\eqno(2.17)
$$
cf. \cite{GZ}. For a given boundary condition bc $\in \{\mbox{per, free} \}$, the
associated {\em Gibbs distribution} in $\Lan$ with parameters $\zbet >0$
is defined by
$$
P_{\nzbet,\bc}(d\omega) = Z_{\nzbet,\bc}^{-1} \, z^{N_n(\omega)}
\exp \left [-\beta H_{n,\bc}(\omega) \right ] \, Q_n(d\omega ), \eqno(2.18)
$$
where
$$
Z_{\nzbet,\bc} = Q_n \big( z^{N_n} \exp [-\beta H_{\nbc}] \big)
\eqno(2.19)
$$
is the so-called partition function. It follows from (2.2) resp. Theorem 1
and (3.3) below that $Z_{\nzbet,\bc}$ is finite. Thus $P_{\nzbet,\bc}$ is
well-defined. Again, we think of it as an element of ${\cal P}$ with support
$\Omega_n$.
\skip\\
{\bf Theorem 2} \enspace {\em Let $\varphi$ be superstable and regular, $\zbet >0$,
and $F : \Pinv \to \R\cup \{ \infty \}$ a measurable functional
satisfying $F \geq -c(1+\rho)$ for some $c < \infty$. Then, for\/} bc $=$
per {\em or\/} free,
$$
\lslog P_{\nzbet,\bc}\big(e^{-\vFR}\big) \leq - \inf [ I_{\zbet} + F_{\lsc} ]
\eqno(2.20)
$$
{\em and}
$$
\lilog P_{\nzbet,\bc}\big(e^{-\vFR}\big) \geq - \inf [ I_{\zbet} + F^{\usc} ],
\eqno(2.21)
$$
{\em where $F_{\lsc}$ is the largest lower semicontinuous minorant and
$F^{\usc}$ the lowest upper semicontinuous majorant of
$F$ relative to $\tau_{\cal L}$. In addition,
$I_{\zbet}$ has $\tau_{\cal L}$-compact level sets, and}
$$
p(\zbet) = \lim_{n \to \infty} \vni \log Z_{\nzbet,\bc}. \eqno(2.22)
$$
\skip
Note that (2.20) and (2.21) take the familiar form of a large deviation
principle when $F$ is chosen to be zero on some measurable set $A \subset
\Pinv$ and equal to $+\infty$ outside $A$. The existence of the limit in (2.22)
is the classical result on the existence of the pressure, cf.
\cite{Ra,RBuch,Rb}. Its
coincidence with (2.15) is called the Gibbs variational formula. The upper
bound (2.20) will be proved in Section 4 and the lower bound (2.21) in Section
5.
\skip
We now turn to large deviations for Gibbs distributions with configurational boundary
conditions, and for infinite-volume Gibbs measures. We need some notations. Let
$C = [-\eh,\eh[^{d} = \Lambda_0$ be the centered half-open unit cube and
$L=\Z^d$. The sets $C+i, i \in L$, form a partition of $\R^d$. For
$n \geq 0$ we set $L_n = L \cap \Lan = \{i \in L: |i| \leq n \}$ and
$$
T_n = \sum_{i \in L_n} N_{C+i}^{2}\quad . \eqno(2.23)
$$
For $t >0$ we define
$$
\Omega(t) = \{ T_n \leq t v_n \enspace
\mbox{for all} \enspace n \geq 0 \}. \eqno(2.24)
$$
The configurations in $\Omega^{\star} = \bigcup_{t>0} \Omega(t)$ are called
{\em tempered}. The multidimensional ergodic theorem shows that $P(\Omega
^{\star}) = 1$ for all $P \in \Pinvquadr$.
For each $\zeta \in \Omega^{\star}$ and $n \geq 0$ we let
$$
H_{n,\zeta}(\omega) =H_n(\omega) + \sum_{\xinom \uLan, y \in \zeta \setminus
\Lan} \varphi(y-x) \eqno(2.25)
$$
denote the Hamiltonian in $\Lan$ with {\em boundary condition} $\zeta$. The
associated Gibbs distributions $P_{\nzbet,\zeta}$ are defined by (2.18) with
bc $=\zeta$. By Lemma 4.2, the last sum in (2.25) exists when $\varphi$
is regular. Under the hypothesis (2.26) below, Lemma 6.1 and
the estimates in the proof of Lemma 6.3 even imply that $Z_{\nzbet,\zeta} < \infty$,
so that $P_{\nzbet,\zeta} $ is well-defined.
A measure $P\in {\cal P}$ is called a {\em tempered Gibbs measure} for $\zbet>0$
if $P(\Omega^{\star})=1$ and, for all $n \geq 0$ and measurable functions
$f \geq 0$ on $\Omega$,
$$
P(f)=\int P(d\zeta) \int P_{\nzbet,\zeta}(d\omega) f(\omega \cup (\zeta
\setminus \Lan)).
$$
Note that the identity above is equivalent to the equilibrium equations in
\cite{Rb}. The following theorem provides a uniform large deviation principle
for Gibbs distributions with (uniformly) tempered boundary conditions, and
a large deviation principle for tempered Gibbs measures. Its proof will be
given in Section 6.
\skip\\
{\bf Theorem 3} \enspace {\em Suppose $\varphi$ is regular and strongly repulsive
in the sense that}
$$
\varphi(x) |x|^d \to \infty \qquad \mbox{as} \enspace |x| \to 0. \eqno(2.26)
$$
{\em Also, let $F$ be as in Theorem 2 and $\zbet > 0$.}
(a) \enspace {\em For each $t>0$, we have}
$$
\lslog \sup_{\zeta \in \Omega(t)} P_{\nzbet,\zeta} \big(e^{-\vFR} \big)
\leq - \inf [ \, I_{\zbet} + F_{\lsc} ] \eqno(2.27)
$$
{\em and}
$$
\lilog \inf_{\zeta \in \Omega(t)} P_{\nzbet,\zeta} \big(e^{-\vFR} \big)
\geq - \inf [ \, I_{\zbet} + F^{\usc} \,]. \eqno(2.28)
$$
{\em Moreover, (2.22) holds with} bc $= \zeta$ {\em uniformly for all $\zeta \in
\Omega(t)$.}
(b) \enspace {\em Each tempered Gibbs measure $P$ with parameters $\zbet$ satisfies
a large deviation principle for $R_n$ with rate function $I_{\zbet}$, in that
inequalities (2.20) and (2.21) hold with $P$ instead of $P_{\nzbet,\bc}$.}
\skip
We note that hypothesis (2.26), together with the lower regularity of $\varphi$,
implies that $\varphi$ is superstable; see Proposition 3.2.8 of \cite{RBuch}.
(2.26) holds in particular when $\varphi$ has a hard core. We also note that
an application of the contraction principle to Theorems 2 and 3 leads to
analoguous large deviation principles for the individual empirical fields
$R_n^{\circ}$ in (2.17); see \cite{GZ} for more details.
\skip
Our last result is a version of the equivalence of ensembles. For any
non-degenerate interval $D \subset [0,\infty[$ and real $\eps$ we consider the
{\em microcanonical Gibbs distribution}
\setcounter{equation}{28}
\begin{eqnarray}
\micro &\equiv& Q_n \left (\, \cdot \,| N_n \in v_n D, H_{\nper}
\leq v_n \eps \right ) \nonumber \\
& = & Q_n \left (\, \cdot \,| \rho(R_n) \in D, \Phi(R_n)
\le\eps \right )
\end{eqnarray}
in $\Lan$ with periodic boundary condition. As we will see, the conditioning
event has positive probability for all sufficiciently large $n$ whenever
$\eps>\inf \phi(D)$. Here $\phi:[0,\infty[ \to \R \cup \{ \infty \}$
is defined by
$$
\phi(\nu) = \inf \big\{ \Phi (P): \pinp,\, \rho(P)=\nu,\, I(P)<\infty \big\}.
\eqno(2.30)
$$
Since $\Phi, \rho$ and $I$ are affine, $\phi$ is convex. $\phi$ is
finite on an interval $[0,\nu(\varphi)[$, where $\nu(\varphi) = \infty$
except when $\varphi$ has a hard core, see Lemma 7.1. Intuitively,
$\nu(\varphi)$ is the close-packing density of $\{ \varphi = \infty \}$-balls.
We write $\mbox{acc}_{n \to \infty} P^{(n)} $ for the set of all accumulation
points ( relative to $\tau_{\cal L}$) of a sequence $P^{(n)}$ in ${\cal P}$.
\skip\\
{\bf Theorem 4} \enspace {\em Suppose $\varphi$ is superstable and regular, let
$D \subset [0,\infty[$ be a non-degenerate interval with $\inf D < \nu(\varphi)$,
and $ \eps > \inf \phi(D)$. Then there exists some $\beta \geq 0$ and $z>0$
such that}
$$
\emptyset \not= \parbox [t]{9mm}{ acc\\[-2.3mm]
{\scriptsize $n\!\!\to\!\!\infty$}} \micro
\subset \{\, I_{\zbet} =0 \} \supset \parbox[t]{9mm}{ acc
\\[-2.3mm]{\scriptsize $n\!\!\to\!\!\infty$}}
P_{\nzbet,\per} \not= \emptyset . \eqno(2.31)
$$
{\em In particular, if $\{ I_{\zbet} =0 \}$ consists of a unique element $P_{\zbet}$
then}
$$
\lim_{n \to \infty} \micro = \lim_{n \to \infty} P_{\nzbet,\per} = P_{\zbet}.
$$
{\em Moreover,}
\setcounter{equation}{31}
\begin{eqnarray}
\lim_{n \to \infty}&&\vni \log Q_n\big(N_n\in v_n D, H_{\nper} \leq v_n \eps\big)
\nonumber\\
&& = - \inf \big\{I(P): \pinp , \rho(P) \in D, \Phi(P) \le\eps \big\}.
\end{eqnarray}
{\em If $\eps<\inf \phi(D)$ or $\inf D > \nu(\varphi)$ the limit in (2.32) equals $-\infty$.}
\skip
(2.31) expresses the asymptotic equivalence of microcanonical and grandcanonical
Gibbs distributions. (2.32) is an analogue of the classical existence result
for the thermodynomic entropy density, cf. \cite{RBuch}. The difference here is
that the particle number is not fixed but ranges in a whole interval, and that
we use periodic boundary conditions. In addition, (2.32) provides a microcanonical
Gibbs variational formula. The proof of Theorem 4 will be given in Section 7 which
also contains some additional information on the involved functions. With the
same techniques, one can derive the asymptotic
equivalence of small canonical and grand canonical ensembles. In addition,
an obvious extension of Lemma 7.2 in the spirit of Sections 4 and 5 leads
to a large deviation principle for the empirical fields $R_n$ under the
microcanonical distributions $\micro$. We leave this to the reader.
\section{The energy density}
\setcounter{equation}{0}
In this section we prove Theorem 1. We begin with some basic estimates.
For $\pinp$ we set $\rhoquadr (P) = P(N_C^2)$, where again $C=\Lambda_0$ is the
centered half-open unit cube. Since
$$
N^2_{\Delta} \leq m \sum_{j=1}^m N^2_{\Delta_j} \qquad \mbox{when} \quad
\Delta \subset \bigcup_{j=1}^m \Delta_j , \eqno(3.1)
$$
$\rhoquadr (P) < \infty$ if and only if $P \in \Pinvquadr$, cf. (2.9).
Suppose now we are given a superstable lower regular potential $\varphi$. The
presence of the
purely repulsive part $\varphi^r$ of $\varphi$ implies that, for fixed $n$,
$H_n$ tends to infinity quadratically as $N_n \to \infty$. This is asserted in
our first lemma. Recall the definitions (2.23) and (2.10) of $T_n$ and $\Phi_n$.
\skip\\
{\bf Lemma 3.1} \enspace {\em There exist constants $a>0, \,b<\infty$ such that for
each $n\geq 0$}
$$
H_n \geq a \, T_n - b \, N_n \eqno(3.2)
$$
and
$$
\Phi_n \geq a \, \rhoquadr - b \, \rho . \eqno(3.3)
$$
\skip\\
\proof (3.2) follows from (2.4), (2.2) and (2.3) by partitioning $\Lan$ into
$k^d\, v_n$ half-open cubes $\Delta_j$ of size $1/k$ (where $k$ is the
smallest integer exceeding $1/\delta$ for the number $\delta$ in (2.3)), and
applying (3.1) with $\Delta = C+i,i \in L_n$, and $m=k^d$. (One can take
$a=\delta / 2k^d$. $b$ exceeds the constant in (2.2) by $\delta /2$.) (3.3)
follows from (3.2) by integration. $\Box$
\skip
As a consequence of (3.3), if $P \not\in \Pinvquadr$ then $\Phi_n(P) =\infty$
for all $n \geq 0$ and thus\linebreak $\lim_{n \to \infty} \Phi_n(P) = \infty$. To prove
Theorem 1 we can therefore assume that $P \in \Pinvquadr$. In this case,
$\Phi_n(P)$ admits a convenient description in terms of the Palm measure
$P^{\circ}$ of $P$.
\skip\\
{\bf Lemma 3.2} \enspace {\em For all $P \in \Pinvquadr$ and $n\geq 0$ we have $\Phi_n
(P)=P^{\circ}(f_{\varphi,n})$, where}
$$
f_{\varphi,n}(\omega) = \eh \sum_{0 \not= y \in \omega} \varphi(y) \,
|\Lan \cap (\Lan -y)|/v_n, \qquad \ominOm.
$$
\skip\\
\proof Consider the (measurable) function
$$
f_n(x,\omega) = \eh \sum_{0 \not= y \in \omega} \varphi(y) 1_{\Lan \cap
(\Lan - y)}(x),
$$
$x \in \R^d, \ominOm $. Since $\varphi \geq -2b$, $f_n(x,\omega) \geq
-b 1_{\Lan}(x) N_n(\vartheta_{-x} \omega)$. Also,
$$
\int P^{\circ}(d \omega) \int dx \,\, 1_{\Lan}(x)N_n(\vartheta_{-x}\omega) =
P(N_n^2) < \infty
$$
for all $P \in \Pinvquadr$, by (2.8). Equation (2.8) therefore also holds for
$f=f_n$. But for $f=f_n$, the left-hand side of (2.8) coincides with $\Phi_n(P)$
and the right-hand side with $P^{\circ}(f_{\varphi,n})$. $\Box$
\skip
Since $|\Lan \cap (\Lan -y)|/v_n \to 1$ as $n \to \infty$, it is natural
to expect that $P^{\circ}(f_{\varphi,n}) \to P^{\circ}(f_{\varphi})$, which
will give us the first assertion of Theorem 1. To make this rigorous we need
the lower regularity of $\varphi$. Let again $L=\Z^d$ and, for each
$i \in L$,
$$
\psi_i = \psi(\, d(C,C+i) \,), \eqno(3.4)
$$
where $d(C,C+i)=(|i|-1)_{+}$ is the distance of $C$ and $C+i$. Hypothesis (2.6)
implies that $\sum_{i \in L} \psi_i < \infty$. Moreover,
$$
f_{\varphi} \geq - \\ \eh \sum_{i \in L} \psi_i \, N_{C+i}\, , \eqno(3.5)
$$
and the function on the right-hand side is $P^{\circ}$-integrable for all
$P \in \Pinvquadr$ because for all $i \in L$
\setcounter{equation}{5}
\begin{eqnarray}
P^{\circ}(N_{C+i})&=&\int P(d\omega) \sum_{\xinom \cap C} N_{C+i}(\thx \omega)
\nonumber\\
&\leq& P(N_C N_{\Lambda_1+i}) \leq P(N_C^2)^{1/2} \, P(N^2_{\Lambda_1})^{1/2}\nonumber\\
&\leq& 3^d \, \rhoquadr(P).
\end{eqnarray}
In the last step we used (3.1). It follows that $\Phi(P) \equiv P^{\circ}
(f_{\varphi})$ is well-defined for all $P \in \Pinvquadr$. We now compare
$\Phi(P)$ with $\Phi_n(P)$.
\skip\\
{\bf Lemma 3.3} \enspace {\em There exists a sequence $\epsilon_n \to 0$ such
that for all $P \in \Pinvquadr$ and $n \geq 0$,}
$$
\Phi(P) \geq \Phi_n(P) - \epsilon_n \, \rhoquadr(P).
$$
\skip\\
\proof Let $P \in \Pinvquadr$ and $n \geq 0$ be given. Since $|\Lan \cap(\Lan
-y)|/v_n \leq 1$, (2.5) and Lemma 3.2 imply that $\Phi(P) \geq \Phi_n(P) -
\epsilon_n(P)$, where
$$
\epsilon_n(P) = \eh \int P^{\circ}(d\omega) \sum_{0 \not= y \in \omega}
\psi(y) \, |\Lan \setminus(\Lan -y)|/v_n.
$$
Distinguishing the cubes $C+i$ containing $y$, we conclude from (3.4) and (3.6)
that $\epsilon_n(P) \leq \epsilon_n \, \rhoquadr(P)$, where
$$
\epsilon_n = \eh \, 3^d \sum_{i \in L} \psi_i \max_{y \in C+i}
|\Lan \setminus (\Lan -y)|/v_n.
$$
But the dominated convergence theorem shows that $\epsilon_n \to 0$ as $n \to
\infty$. $\Box$
\skip\\
{\em Proof of Theorem 1.} \enspace We first show that, for all $\pinp$, $\lim_{n \to \infty}
\Phi_n(P)$ exists and has the claimed value. The case $P \not\in \Pinvquadr$
was already discussed after Lemma 3.1. For $P \in \Pinvquadr$, Lemma 3.2 and
Fatou's lemma imply that
$$
\Phi(P)=P(\liminf_{n \to \infty} f_{\varphi,n}) \leq \liminf_{n \to \infty}
\Phi_n(P)
$$
because the functions $f_{\varphi,n}$ are not less than the right-hand side
of (3.5). Together with Lemma 3.3, this shows that $\Phi(P)=\lim_{n \to \infty}
\Phi_n(P)$.
To prove the lower semicontinuity of $\Phi$ we note first that each $\Phi_n$
is lower semicontinuous. This is because $H_n$ satisfies (2.2) and is thus
the supremum of functions in ${\cal L}$. Now let $c \in \R$ and
$(P_{\alpha})_{\alpha \in D}$ be a net in $\{\Phi \leq c\}$ which converges
(in $\taul$) to some $\pinp$. Then
$\rho(P_{\alpha}) \to \rho(P) < \infty$. We thus can assume without loss that
$s \equiv \sup_{\alpha} \rho(P_{\alpha}) < \infty$. In view of (3.3) and
the first part of this proof, we have for all $\alpha \in D$
$$
c \geq \Phi(P_{\alpha}) \geq a \, \rhoquadr(P_{\alpha}) -
b \, \rho(P_{\alpha})
$$
and thus $\rhoquadr(P_{\alpha}) \leq (c+bs)/a \equiv c'$. Together with
Lemma 3.3 this implies that
\begin{eqnarray*}
\Phi_n(P)-\epsilon_n c'&\leq& \liminf_{\alpha \in D} \Phi_n(P_{\alpha})-
\epsilon_n c' \\ & \leq& \liminf_{\alpha \in D} \Phi(P_{\alpha}) \leq c
\end{eqnarray*}
for all $n \geq 0$. Letting $n \to \infty$ we see that $P \in \{\Phi \leq
c \}$. $\Box$
\skip
We conclude this section proving the compactness of the level sets of the
functionals $I_{\zbet}$ defined in (2.14).
\skip\\
{\bf Lemma 3.4} \enspace {\em For any two numbers $c_1, c_2\,$, the set
$\left\{ I+ \Phi \leq c_1 + c_2 \, \rho \right\}$
is $\taul$-compact.}
\skip\\
\proof The set above is closed because $\rho$ is continuous and $I$ and $\Phi$
are lower semicontinuous. In fact, $I$ even has compact level sets, see
Proposition 2.6 of \cite{GZ}. The same is true for $I^z$, the relative entropy
density with reference measure $Q^z$, the Poisson point random field with
intensity $\rho(Q^z)=z>0$. It is easy to see that $I^z =I_{z,0} = I -\rho
\log z +z-1$. By (2.2), $\Phi \geq - b \rho$. The set under consideration is
therefore contained in the compact set $\{I^z \leq c_1+z-1\}$, where
$z=\exp(b+c_2)$. $\Box$
\section{The upper estimate}
\setcounter{equation}{0}
Let $\varphi$ be superstable and regular and $F:\Pinv \to \R \cup \{
\infty \}$ a measurable function such that $F \geq -c(1+\rho)$ for some
$c<\infty$. In this section we shall prove the following result.
\skip\\
{\bf Proposition 4.1} {\em For\/} bc $=$ per {\em or\/} free,
$$
\lslog Q_n\big (\exp[-\vFR - H_{\nbc}]\big)
\leq - \inf [I+\Phi +F_{\lsc}].
$$
\skip
In the case bc $=$ per, the above proposition follows directly from the results of
\cite{GZ} combined with those of Section 3. Namely, let $G=F+\Phi$. The
integrand under consideration then equals $\exp[-v_n \,G(R_n)]$. By Theorem 1
and (3.3), $G \geq -c'(1+\rho )$ with $c'=c+b$, and $G$ is clearly measurable
because so is $\Phi$. Theorem 3.1 of \cite{GZ} thus implies that the $\limsup$
in Proposition 4.1 is not larger than $-\inf[I+G_{\lsc}]$. But $G_{\lsc} \geq
F_{\lsc}+\Phi$ because $\Phi$ is lower semicontinuous.
The proof of Proposition 4.1 for bc $=$ free is based on a comparison
with the case of periodic boundary conditions. We need several lemmas. First,
we use the regularity of $\varphi$ to estimate the interaction of suitably
separated configurations.
For each integer $k \geq 0$ we define
$$
\delta_k = \eh \sum_{\ell \geq k} \partial \psi(\ell)\, v_{\ell}\, ,\eqno(4.1)
$$
where $\partial \psi(\ell) = \psi(\ell -1)-\psi(\ell) \geq 0$ for $\ell \geq 1,
\,\partial \psi(0)=0$, and $\psi$ is as in (2.5) and (2.7). By (3.4),
$$
\psi_i = \sum_{\ell \geq |i|} \partial \psi(\ell) \eqno(4.2)
$$
for all $i \in L$. Hence
$$
\sum_{\ell \geq 0} \partial \psi(\ell) \, v_{\ell} = \sum_{i \in L} \psi_i <\infty
$$
and thus $\delta_k \to 0$ as $k \to \infty$. For all $n \geq 0,\,\ominOm_n$ and
$\zeta \in \Omega$ we have
$$
\sum_{\xinom,y \in \zeta} \psi(y-x) \leq \sum_{i \in L_n,j\in L} \psi_{i-j}
\, N_{C+i}(\omega) \, N_{C+j}(\zeta). \eqno(4.3)
$$
We estimate the long-distance contribution to the right-hand side of (4.3) in
two cases. (The second case will be used later in Section 6.)
\skip\\
{\bf Lemma 4.2} \enspace {\em For all $n \geq 0,\,\ominOm_n $ and $k \geq 0$,}
\begin{eqnarray*}
\sum_{i \in L_n,j \in L:|i-j|\geq k} &\psi_{i-j}& \, N_{C+i}(\omega) \,
N_{C+j}(\zeta) \\ & \le&
\left\{ \begin{array}{ll}
\delta_k(1+2^d)\, T_n(\omega)& \mbox{if $\zeta = \omega^{(m)}$ for some
$m \geq n$,}\smallskip\\
\delta_k \left[ T_n(\omega)+v_n \, t \, 2^d \right] & \mbox{if $\zeta \in
\Omega(t)$ for some $t>0.$} \end{array} \right.
\end{eqnarray*}
\skip\\
\proof In view of (4.2), the sum above is not larger than
$$
\sum_{\ell \geq k} \partial \psi(\ell) \sum_{i \in L_n, j \in L_{\ell}+i}
N_{C+i}(\omega)N_{C+j}(\zeta).
$$
Using the inequality $uv \leq (u^2+v^2)/2$ we obtain the upper bound
$$
\eh \sum_{\ell \geq k} \partial \psi(\ell) \left[ v_{\ell}\, T_n(\omega)+
S_{n,\ell}(\zeta) \right]
$$
with
$$
S_{n,\ell}(\zeta ) = \sum_{j \in L} N_{C+j}(\zeta)^2 \,\mbox{ card }
\big(L_n \cap(L_{\ell} +j)\big).
$$
For $\zeta = \omega^{(m)}$ with $m \geq n$ we have
\begin{eqnarray*}
S_{n,\ell}(\omega^{(m)})& =&\sum_{j'\in L_n} N_{C+j'}(\omega)^2
\sum_{i \in L_n} \mbox{card} \{j\in L_{\ell}+i:j\equiv j'\,\mbox{ mod }\, 2m+1\}
\\
&\leq& v_n \, (2^d v_{\ell}/v_m)\, T_n(\omega) \leq 2^d \, v_{\ell} \, T_n(
\omega).
\end{eqnarray*}
On the other hand, if $\zeta \in \Omega(t)$ then
\begin{eqnarray*}
S_{n,\ell}(\zeta) &\leq& (v_n \wedge v_{\ell}) \, T_{n+\ell}(\zeta)\\
&\leq& v_{n \wedge \ell} \,\, t \,\, v_{2(n \vee \ell)} \leq t \, 2^d \, v_n \,
v_{\ell}
\end{eqnarray*}
which implies the lemma in the second case. $\Box$
\skip
Recall the notation $r(\varphi)$ in (2.7).
\skip\\
{\bf Lemma 4.3} \enspace {\em For all $n \geq 0$ and $k \geq r(\varphi)$,}
$\vert H_n - H_{n+k,\per}\vert\le 2^d\delta_k T_n \quad \mbox{{\em on\/ }}
\Omega_n.$
\skip\\
\proof After a comparison of (2.1) and (2.16), the lemma follows
immediately from (2.5) and (2.7) together with (4.3) and Lemma 4.2. $\Box$
\skip
Next we need to compare $R_n$ with $R_{n+k}$. For $n \ge 0, s>0$ we define
$$
\Omega(s,n) = \left\{ \ominOm: T_n(\omega) \le s \, v_n \right\}.\eqno(4.4)
$$
\skip\\
{\bf Lemma 4.4} \enspace {\em For all $k \ge 1,\, s>0$ and $f \in {\cal L}$,}
$$
\lim_{n\to \infty} \sup_{\ominOm(s,n+k)} \left\vert R_{n+k,\omega}(f)-
R_{n,\omega}(f) \right\vert =0.
$$
\skip\\
\proof The case of bounded $f$ is trivial. We thus assume for
simplicity that $|f|\le N_{\Delta}$ for some centered cube $\Delta \supset
C$ with $f=f(\cdot \cap \Delta)$. For each $n$ we can write, setting $m=n+k$,
\begin{eqnarray*}
\big\vert R_{m,\omega}(f)-R_{n,\omega}(f) \big\vert &\le&
(\vni -v_m^{-1}) \int_{\Lambda_m} N_{\Delta +x}(\omega^{(m)}dx \\
&&+\,\vni \int_{\Lambda_m \setminus \Lan}N_{\Delta +x}(\omega^{(m)})dx \\
&&+\,\vni \int_{\partial \Lan} \left( N_{\Delta +x}(\omega^{(m)})+
N_{\Delta +x}(\omega^{(n)})\right) \, dx,
\end{eqnarray*}
where $\partial \Lan = \{x \in \Lan : x+\Delta \not\subset \Lan \}$. On the other
hand, for each $m$ with $\Lambda_m \supset \Delta +C$ and all $V \subset
\Lambda_m$ we have
\begin{eqnarray*}
\int_V N_{\Delta +x}(\omega^{(m)})dx &\le&
\sum_{i \in L} N_{C+i}(\omega^{(m)}) \, | \{x \in V:(\Delta +x) \cap (C+i)
\ne \emptyset \} | \\
&\le& 2^d \, |\Delta|\, \sum_{i \in L \cap(V+\Delta +C)} N_{C+i}(\omega^{(m)})\\
&\le& 4^d \, |\Delta|\, \sum_{j \in L_m(V)} N_{C+j}(\omega)\\
&\le& 4^d \, |\Delta| \, \mbox{card}L_m(V)^{1/2} \, T_m(\omega)^{1/2}
\end{eqnarray*}
with $L_m(V)=\{j \in L_m:j \equiv i\,\mbox{ mod }\,2m+1\,\mbox{ for some }\,i \in V+
\Delta +C \}$. In the cases $V=\Lambda_m, V=\Lambda_m \setminus \Lan$, and $V
= \partial \Lan$ we have $\mbox{card}L_m(V) \le c_{\Delta} |V|$ for some
constant $c_{\Delta}<\infty$. Combining all estimates above one can now
easily complete the proof. $\Box$
\skip\\
{\bf Lemma 4.5} \enspace {\em For any three constants $c_1, c_2, c_3>0$ there
exists some $s=s(c_1,c_2,c_3)$\\$<\infty$ such that
$$
Q_n \big( e^{c_1N_n-c_2T_n};\Omega(s,n)^c\big)\le e^{-c_3v_n}
$$
for all $n\ge 0$.}
\skip\\
\proof Let $s$ be so large that $sc_2\ge c_3+e^{c_1}-1$. The result
then follows from the inequality
$$
1_{\Omega(s,n)^c}\le \exp[c_2(T_n-sv_n)]
$$
and the fact that $Q_n(e^{c_1N_n})=\exp[(e^{c_1}-1)v_n].\enspace \Box$
\skip
Before completing the proof of Proposition 4.1 we need one further notation.
We let ${\cal U}$ denote the system of all sets of the form
$$
U=\left\{(P_1,P_2)\in \Pinv \times \Pinv: \max_{1\le i\le k}
|P_1(f_i)-P_2(f_i)|\le \epsilon \right\}
$$
with $k\ge 1,\epsilon>0,$ and $f_1,\dots,f_k \in {\cal L}.$ By definition,
${\cal U}$ is a uniformity base for $\taul$.
\skip\\
{\em Proof of Proposition 4.1 in the case bc $=$ free}. \enspace For given
$s>0$ and arbitrary $n$ we can write
$$
Q_n\big(\exp[-\vFR-H_n]\big)=p_n(s)+q_n(s),
$$
where $p_n(s)$ is defined by restricting the integral on the left-hand side
to the set $\Omega(s,n)$ and $q_n(s)$ is the remaining contribution corresponding
to $\Omega(s,n)^c$. Let $\tau>0$ be any given number, and let $s=s(c+b,a,\tau
+c)$ be chosen according to Lemma 4.5. Here $c$ is the constant appearing in
the hypothesis $F\ge-c(1+\rho)$, and $a,\,b$ are as in (3.2). It then follows
that $q_n(s)\le\exp[-\tau v_n]$ for all $n$.
To estimate the main term $p_n(s)$ we choose an arbitrary integer $k\ge r(
\varphi)$. Then for each $n\ge0$ we can write, setting again $m=n+k$ and
using Lemma 4.3 and (2.16),
\begin{eqnarray*}
p_n(s)&=&e^{v_m-v_n} \, Q_m\Big(\exp[-\vFR-H_n];\Omega(s,n)\cap
\{ N_{\Lambda_m \setminus \Lan}=0\} \Big)\\
&\le& \exp[v_m-v_n+2^d\delta_k\,s\,v_n]\, Q_m\big(\exp[-G_m]\big),
\end{eqnarray*}
where $G_m=\vFR+v_m\Phi(R_m)+\infty\cdot 1_{\Omega(s,m)^c}\,$. Next we choose any
$U\in{\cal U}$ and $\ell>0$ and define $F^{\ell}=F\wedge \ell$,
$$
F^{\ell}_U(P)=\inf\{F^{\ell}(P'):(P,P')\in U\},
$$
and $G_U^{\ell}=F_U^{\ell}+\Phi$. Lemma 4.4 asserts that $(R_m,R_n)\in U$
on $\Omega(s,m)$ for sufficiently large $m$. Since $v_nF\ge v_mF^{\ell}
-(v_m-v_n)\ell$, we conclude that for these $m$
$$
v_m^{-1}\,G_m\ge G_U^{\ell}(R_m)-(1-v_n/v_m)\ell.
$$
But this is all what is needed for the large deviation upper bound (3.2) of
\cite{GZ}, cf. the proof of Lemma 5.6 there. (Measurability of $G_U^{\ell}$
is not required.) Hence
$$
\limsup_{n\to\infty}v_m^{-1}\log Q_m(\exp[-G_m]) \le-\inf[I+(G_U^{\ell})_{\lsc}].
$$
In view of the lower semicontinuity of $\Phi$, $(G_U^{\ell})_{\lsc}\ge
(F_U^{\ell})_{\lsc}+\Phi$. Also, Lemma 3.4 shows that the argument of Remark 1.4
of \cite{G} can be applied, yielding
$$
\sup_{\ell>0,U\in{\cal U}} \inf [I+\Phi+(F_U^{\ell})_{\lsc}]=
\inf[I+\Phi+F_{\lsc}] \equiv \gamma.
$$
Hence
$$
\lslog p_n(s) \le 2^d\delta_k s-\gamma.
$$
Since $k$ is arbitrary, we finally get
$$
\lslog [p_n(s)+q_n(s)]\le-\gamma \wedge\tau,
$$
and letting $\tau\to\infty$ we obtain the result. $\Box$
\section{The lower estimate}
We still assume that $\varphi$ is superstable and regular. Our proof of the
lower bound
(2.21) follows the standard device of changing the measure so that untypical
events become typical, and controlling the Radon-Nikodym density by means of
McMillan's theorem. But some refinements are necessary. The basic observation
is that the familiar approximation of invariant by ergodic probability measures
can be sharpened as follows. For $q>0$ we define
\be
\Gamma_q=\Big\{\ominOm:\varphi(x-y)\le q\, \mbox{ and }\, |x-y|\ge 1/q
\,\mbox{ for any two distinct }\, x,y \in\omega\Big\}.
\ee
It is easy to check that $\Gamma_q$ is measurable.
\skip\\
{\bf Lemma 5.1} \enspace {\em Let $\pinp$ be such that $I(P)+\Phi(P)<\infty$. For
each open neighbourhood $U$ of $P$ and any $\epsilon>0$ there exists some
$\Theta$-ergodic $P'\in U$ such that $I(P')*0$, $\Phi$ is continuous on ${\cal P}_
{\Theta,q}$.}
\skip\\
\proof For each $n\ge0$, $H_n$ is local and bounded on $\Gamma_q$. Hence
$\Phi_n$ is continuous on ${\cal P}_{\Theta,q}$. Also, a glance at Lemma 3.3
and its proof shows that the convergence $\Phi_n\to\Phi$ is uniform on
${\cal P}_{\Theta,q}$. This gives the result. $\Box$
\skip
Let $r\ge r(\varphi)$ be a fixed integer. For $n\ge r$ we define the modified
empirical fields
\be
R_{n,\omega}^{\#}=R_{n,\omega\cap\Lambda_{n-r}}.
\ee
\skip\\
{\bf Lemma 5.3}\enspace {\em For each ergodic $\pinp, \,R_n^{\#}\to P$ in
$P$-probability as $n\ti$.}
\skip\\
\proof This follows in the same way as (1.2); cf. the proof of Remark 2.4 in
\cite{GZ}. $\Box$
\skip
Let $F$ be as in Theorem 2.
\skip\\
{\bf Proposition 5.4} \enspace {\em For all $\pinp$ and} bc $\in \{$per, free$\}$,
$$
\lilog Q_n\big(\exp\left[-\vFR-H_{\nbc}\right]\big)\\
\ge -\big[I(P)+\Phi(P)+F^{\usc}(P)\big].
$$
\skip\\
\proof We may assume without loss that the right-hand side of the asserted
inequality, denoted by $-\gamma_P$, is finite. By Lemma 5.1, we can even assume
that $P$ is ergodic and supported on $\Gamma=\Gamma_q$ for some $q\ge\psi(r)$.
For $n\ge r$ let $P_n^{\#}=P_{n-r}$. We think of $P_n^{\#}$ as a measure
on $\Omega_n$ which leaves $D_n=\Lan\setminus\Lambda_{n-r}$ free of particles.
Since $I(P)<\infty, \,P_{n-r}\ll Q_{n-r}$ with a density $f_{n-r}$. Hence
$P_n^{\#}\ll Q_n$ with the density $f_n^{\#}=1_{\{N_{D_n}=0\}}f_{n-r}
\exp[v_n-v_{n-r}]$. Given any $\epsilon>0$, we define
$$
A_n=\left\{F(R_n)0,\,t>0$ be given. If $n$ is sufficiently
large,}
$$
\inf_{\zeta\in\Omega(t)}H_{n,\zeta}\ge H_n-\epsilon\,T_n^h-\epsilon\,v_n,
$$
{\em where $h$ is as in Lemma 6.1.}
\skip\\
\proof In view of (2.25), (2.5) and (4.3) we must show that
\be
\sup_{\zeta\in\Omega(t)}\sum_{i\in L_n,j\not\in L_n}\psi_{i-j}\,N_{C+i}\,
N_{C+j}(\zeta)\le\epsilon\, T_n^h+\epsilon\,v_n
\ee
for large $n$. Let $q$ be such that $h(\ell)\ge\ell^2$ for all $\ell\ge q$. Then
$T_n\le q^2\,v_n+T_n^h$ for all $n$. By Lemma 4.2, the contribution of all $i,j$
with $|i-j|\ge k$ to the sum on the left-hand side of (6.3) is not larger than
$\delta_k\,[T_n^h+v_n(t2^d+q^2)]$. Choosing $k$ sufficiently large we can achieve
that this is less than $\epsilon(T_n^h+v_n)/2$.
To estimate the remaining part of the sum in (6.3) we define $g(\ell)=\ell\,
h(\ell)^{1/2},\,\ell\ge0$. $g$ is increasing with $g(0)=0$ and satisfies
$g(\ell)/\ell^2\ti$ and $g(\ell)/h(\ell)\to0$ as $\ell\ti$. Let $g^*$ be as in
Lemma 6.2. Then $\ell m \le g(\ell)+g^*(m)$ for all $\ell,m\ge0$ and therefore
\begin{eqnarray}
\sum_{i\in L_n,j\not\in L_n:|i-j|0$. Together with Proposition 4.1 ( for bc $=$ free ), this gives
the estimate
$$
\lslog \bar p_n(t)\le(\epsilon+\epsilon s-\gamma)\vee(-\tau),
$$
where $\gamma = \inf [I+\Phi+F_{\lsc}]$. Letting first $\epsilon\to0$ and then
$\tau\ti$ we obtain the uniform upper bound
\be
\lslog\bar p_n(t)\le-\gamma.
\ee
To obtain a lower estimate of
$$
\underline{p}_n(t)\equiv\inf_{\zeta\in\Omega(t)} Q_n\big(\exp[-\vFR-
H_{n,\zeta}]\big)
$$
we can proceed just as in the proof of Proposition 5.4, the only difference
being that now we must show that
$$
\inf_{\zeta\in\Omega(t)}P_{n-r}\big(\vni\, H_{n,\zeta}<\Phi(P)+\epsilon\big)
\to1 \qquad\mbox{as}\quad n\ti
$$
for any given $\epsilon>0$, each ergodic $\pinp$ which is supported on some
$\Gamma_q$, and suitable $r\ge r(\varphi)$. But (2.7), (4.3) and Lemma 4.2
imply that
$$
\sup_{\zeta\in\Omega(t)}H_{n,\zeta}\le H_n+\epsilon\,v_n/2\qquad\mbox{on}
\quad \Gamma_q\cap\Omega_{n-r}
$$
when $r$ is large enough, and we know from the proof of Proposition 5.4
that
$$
P_{n-r}\big(\vni H_n<\Phi(P)+\epsilon/2\big)\to1\qquad\mbox{ as }n\ti.
$$
We thus arrive at the uniform lower bound
\be
\lilog\underline{p}_n(t)\ge-\inf[I+\Phi+F^{\usc}].
\ee
Assertion (a) of Theorem 3 now follows from (6.5) and (6.6) in the same way as
Theorem 2 from Propositions 4.1 and 5.4. $\Box$
\skip
We now turn to the large deviation principle for tempered Gibbs measures. Our
main tool are the remarkable probability estimates of Ruelle \cite{Rb}.
The implication (a)$\Rightarrow$(b) of his Corollary 5.3 gives us the following.
\skip\\
{\bf Proposition 6.4}\enspace {\em For given $\zbet>0$, there exist constants
$\gamma,\delta>0$ such that, for all tempered Gibbs measures $P$ with parameters
$\zbet$ and all $n\ge0$, $P_n$ is absolutely continuous relative to $Q_n$ with
a density $f_n$ satisfying} $f_n\le\exp[-\gamma T_n+\delta N_n].$
\skip
For $t>0$ and $n\ge0$ we define
\be
\Omega_n(t)=\left\{\zeta\in\Omega:\zeta\setminus\Lan\in\Omega(t)\right\}.
\ee
\skip\\
{\bf Corollary 6.5}\enspace {\em Let $\zbet>0$ and $\tau,c>0$ be any constants. Then
there exists some $t>0$ such that, for each tempered Gibbs measure $P$ with
parameters $\zbet$ and all $n\ge0$, }
$$
P\big(e^{cN_n};\Omega_n(t)^c\big)\le e^{-\tau v_n}\, .
$$
\skip\\
\proof We may assume that $\tau>\log 2$. Let $\gamma,\delta$ be as in Proposition 6.4 and
$t$ so large that $t\,\gamma-e^{c+\delta}+1\ge\tau$. Then we can write
\bea
P\big(e^{cN_n}\,;\,\Omega_n(t)^c\big)&\le&\sum_{\ell>n}P\big(e^{cN_n};
T_{\ell}>t\,v_{\ell}\big)\\
&\le&\sum_{\ell>n}P\big(\exp[c\,N_{\ell}+\gamma\,T_{\ell}-\gamma\,t\,v_{\ell}]\big)\\
&\le&\sum_{\ell>n}Q\big(\exp[(c+\delta)N_{\ell}-\gamma\,t\,v_{\ell}]\big)\\
&\le&\sum_{\ell>n}\exp[-\tau\,v_{\ell}]\, .
\eea
This implies the corollary because $v_{\ell}>v_n+\ell-n$ when $\ell>n$. $\Box$
\skip\\
{\em Proof of Theorem 3, assertion (b).}\enspace Let $\zbet>0$ and $P$ be a
tempered Gibbs measure for $\zbet$. Also, let $F:\Pinv\to \R\cup\{\infty\}$
be such that $F\ge-c(1+\rho)$ for some $c<\infty$, $\tau>c$ an arbitrary constant,
and $t$ as in Corollary 6.5. Then
$$
P\big(e^{\vFR};\Omega_n(t)^c\big)\le e^{-(\tau-c)v_n}
$$
and
\bea
P\big(e^{\vFR};\Omega_n(t)\big)&=&\int_{\Omega_n(t)}P(d\zeta)\,
P_{\nzbet,\zeta}\big(e^{\vFR}\big)\\
&\le&\sup_{\zeta\in\Omega(t)}P_{\nzbet,\zeta}\big(e^{\vFR}\big)
\eea
for all $n$. The uniform upper bound (2.27) thus implies
$$
\lslog P\big(e^{\vFR}\big)\le-\inf[I_{\zbet}+F_{\lsc}]\wedge(\tau-c),
$$
and letting $\tau\ti$ we obtain the upper large deviation bound for $P$. The
lower bound follows from the inequality
$$
P\big(e^{\vFR}\big)\ge P(\Omega(t))\,\inf_{\zeta\in\Omega(t)}
P_{\nzbet,\zeta}\big(e^{\vFR}\big)
$$
together with (2.28) and the fact that $P(\Omega(t))>0$ for sufficiently large $t$.
$\Box$
\section{The equivalence of ensembles}
\setcounter{equation}{0}
In this section we prove Theorem 4. Let $\varphi$ be superstable and regular.
We first look at the function $\phi$ defined in (2.30). Let
$$
\Omega_{\varphi}=\Big\{\ominOm:\varphi(x-y)<\infty\;\mbox{for all } x,y\in\omega,
x\ne y\Big\}
$$
and
\be
\nu(\varphi)=\sup\Big\{\rho(P):\pinp, I(P)<\infty, P(\Omega_{\varphi})=1\Big\}.
\ee
Clearly, $\nu(\varphi)$ depends on $\varphi$ only via the set $\{\varphi=\infty\}$.
If $\varphi$ is finite (except possibly at the origin) then $\nu(\varphi)=\infty$.
For in this case we have $\Omega_{\varphi}=\Omega$ so that the Poisson point random
fields $P=Q^z$ of arbitrarily large intensity $z$ appear on the right-hand side
of (7.1).
\skip\\
{\bf Lemma 7.1}\enspace {\em $\phi$ is convex. On $[0,\nu(\varphi)[$, $\phi$ is
finite and continuous. Also, $\phi(\nu)\ge a\,\nu^2-b\,\nu$ for all $\nu\ge0$
and the constants $a,b$ in (3.3), and if $\lambda\equiv\int\varphi(x)dx$
is finite then $\phi(\nu)\le\lambda\,\nu^2/2$ for all $\nu\ge0$. (In particular,
it follows that $\lambda\ge a$.)}
\skip\\
\proof The inequalities for $\phi$ follow from (3.3) and the easily verified fact
that $\Phi(Q^z)=\lambda\, z^2/2$ for all $z>0$. It only remains to show that
$\phi$ is finite on $[0,\nu(\varphi)[$. For, together with the obvious convexity
of $\phi$, this will imply that $\phi$ is continuous on $]0,\nu(\varphi)[$. The
continuity of $\phi$ at $0$ is clear because $\phi(0)=0$ and $\phi(\nu)\ge-b\,
\nu$.
To prove the finiteness of $\phi$ we fix any $\nu<\nu(\varphi)$. By (7.1) there
exists some $\pinp$ such that $\rho(P)>\nu, I(P)<\infty$, and $P(\Omega_{\varphi})
=1$. A glance at the proof of Lemma 5.1 shows that its hypothesis $\Phi(P) <
\infty$ can be replaced by the condition $P(\Omega_{\varphi})=1$. Therefore we can find
some $P'\in\Pinv$ such that $\rho(P')>\nu, I(P')<\infty$, and $P'(\Gamma_q)=1$
for some $q<\infty$. Since $\Gamma_q$ is $\Theta$-invariant, the Palm measure
of $P'$ is also supported on $\Gamma_q$. This and the regularity of $\varphi$
immediately imply that $\Phi(P')<\infty$. Writing $s=\nu/\rho(P')$ we thus
obtain that $P''\equiv sP'+(1-s)\delta_{\emptyset}$ has the properties
$\rho(P'')=\nu, \Phi(P'')<\infty$, and $I(P'')<\infty$. Hence $\phi(\nu)<\infty$.
$\Box$
\skip
In view of (7.3) and (7.7) below, the function $\phi(\nu)$ coincides with the
function $\eps_0(\rho)$ on page 50 of \cite{RBuch}. Note, however, that not
necessarily $\phi(\nu)\ti$ when $\nu\to\nu(\varphi)<\infty$; indeed, if $\varphi$
is a pure hard core potential (taking only the values $0$ and $\infty$) then
$\phi(\nu)=0$ for all $\nu<\nu(\varphi)$.
Next we define
\be
s(\nu,\eps)=-\inf\big\{I(P):\pinp,\, \rho(P)=\nu,\, \Phi(P)\le\eps\big\}.
\ee
Clearly, $s(\,\cdot\, ,\,\cdot\,)$ is increasing in $\eps$ and concave, and (2.30)
means that
\be
\phi(\nu) = \inf \{s(\nu,\,\cdot\,)>-\infty\} \qquad \mbox{for all }\nu\ge0.
\ee
Using the lower semicontinuity of $\Phi$ and the fact that $I$ has compact level sets
\cite{GZ} we also see that $s(\,\cdot\, ,\,\cdot\,)$ is upper semicontinuous.
$s(\,\cdot\, ,\,\cdot\,)$ is the entropy density, as is shown in the next lemma
which proves (2.32).
\skip\\
{\bf Lemma 7.2}\enspace {\em Let $D\subset[0,\infty[$ be a nondegenerate interval
with $\inf D<\nu(\varphi)$, and let $\eps>\inf\phi(D)$. Then}
$$
\lim_{n\ti}\vni \log Q_n\big(N_n\in v_nD, H_{\nper}\le v_n\eps\big)=s(D,\eps)\equiv
\sup_{\nu\in D} s(\nu,\eps).
$$
{\em If $\,\inf D>\nu(\varphi)$ or $\eps<\inf\phi(D)$ then the limit above equals
$-\infty$.}
\skip\\
\proof Let $\bar D$ be the closure and $D^o$ the interior of $D$. By the
continuity of $\phi$ on $[0,\nu(\varphi)[$, $\,\inf \phi(\bar D)=\inf\phi(D^o)
=\inf\phi(D)$. In view of the continuity of $\rho$ and the lower semicontinuity
of $\Phi$, the set $A=\{\rho\in\bar D,\Phi\le\eps\}$ is closed. The upper
bound (2.20) for $\beta=0, z=1$ thus implies that
\be
\lslog Q_n(R_n\in A)\le-\min I(A)
\ee
($I$ attains its minimum over $A$ because $I$ has compact level sets.) On the
other hand, consider the convex set $U=\{\rho\in D^o,\Phi<\eps\}$. Since
$\eps>\inf\phi(D^o)$, $U\cap\{I<\infty\}\ne\emptyset$. A standard convexity
argument (together with the fact that $I$, $\rho$ and $\Phi$ are affine)
thus shows that $\inf I(U)=\inf I(A)=s(D,\eps)$. But
$$
\lilog Q_n(R_n\in U)\ge-\inf I(U)
$$
by the arguments of Section 5. Indeed, let $P\in U$ be such that $I(P)<\infty$.
For given $\epsilon>0$, Lemma 5.1 provides us with some ergodic $P'\in U$ such that
$I(P')**0$. As in the proof of Proposition 5.4,
we have the estimate
$$
\vni\log Q_n(R_n\in U)\ge-I(P')-\epsilon+\vni\log P'\big(R_n^{\#}\in U,
\,\vni\log f_n^{\#}\le I(P')+\epsilon\big),
$$
and McMillan's theorem and Lemmas 5.2 and 5.3 show that the last probability
tends to $1$ as $n\ti$. The second statement of the lemma follows from (7.3)
and (7.4). $\Box$
\skip\\
{\bf Proposition 7.3}\enspace {\em Suppose $D\subset[0,\infty[$ is a nondegenerate
interval with $\inf D<\nu(\varphi)$, and let $\eps>\inf\phi(D)$. Then the sequence
$(\micro)_{n\ge0}$ is relatively compact \linebreak(in $\taul$), and every accumulation point
belongs to the set $M_{D,\eps}$ of all $I$-minimizers in $\{\rho\in\bar D,
\Phi\le\eps\}$.}
\skip\\
\proof Lemma 7.2 implies that the conditioning event in (2.29) has positive
probability for sufficiently large $n$. Thus, for all these $n$, $\micro$ is
well-defined. Let $\hat P^{(n)}\in {\cal P}$ denote the measure relative to
which the configurations in the disjoint blocks $\Lan +(2n+1)i,\linebreak i\in L$, are
independent with identical distribution $\micro$, and $P^{(n)}\in\Pinv$ the
invariant average of $\hat P^{(n)}\circ\thx^{-1},\, x\in\Lan$. Since
$\micro$ is invariant under translations modulo $\Lan$, $P^{(n)}(f)-
\micro(f)\to0$ as $n\ti$, for all $f\in{\cal L}$; cf. Lemma 4.6 of \cite{G}.
Just as in Step 3) of the proof of Lemma 5.1 we obtain
$$
I(P^{(n)})\le\vni\,I(\micro;Q_n)=-\vni\log Q_n\big(\rho(R_n)\in D,\Phi(R_n)
\le\eps\big).
$$
Lemma 7.2 thus implies that
$$
\limsup_{n\ti} I(P^{(n)})\le-s(D,\eps)<\infty.
$$
Since $I$ has compact (and sequentially compact) level sets, it follows that
the sequence $(P^{(n)})_{n\ge0}$,
and thus also the asymptotically equivalent sequence $(\micro)_{n\ge0}$, are
relatively compact. Moreover, each accumulation point of any of these sequences belongs
to the set $\{I\le-s(D,\eps)\}$. On the other hand, Lemma 5.7 of \cite{GZ}
asserts that $P^{(n)}$ is asymptotically equivalent to
$$
\micro R_n \equiv\int\micro(d\omega)\, R_{n,\omega}
$$
as $n\ti$, and the latter measures all belong to the closed convex set
$\{\rho\in\bar D,\Phi\le\eps\}$, as is easily seen by approximating $\micro$
by suitable discrete measures. This proves the proposition. $\Box$
\skip
To deduce the first part of (2.31) from Proposition 7.3 we choose a tangent plane
$(\nu,\eps')\to p+\beta\eps'-\nu\log z$ to the concave function $s(\,\cdot\, , \,
\cdot\,)$ on the subset of $\bar D\times]-\infty,\eps]$ on which $s(\,\cdot\, , \,
\cdot\,)$ attains its maximum $s(D,\eps)$. The monotonicity of $s(\nu,\,\cdot\,)$
implies that $\beta\ge0$. It then follows that, for all $\pinp$ with $\Phi(P)<\infty$,
$$
I(P)\ge-s(\rho(P),\Phi(P))\ge-p-\beta\Phi(P)+\rho(P)\log z
$$
with equality when $P\in M_{D,\eps}$. Since $M_{D,\eps}\ne\emptyset$, we may conclude
that $p=p(\zbet)$ and
\be
M_{D,\eps}\subset\{I_{\zbet}=0\} .
\ee
Together with Proposition 7.3, this gives the first part of (2.31). The second part
is the subject of the next proposition.
\skip\\
{\bf Proposition 7.4}\enspace {\em For all $\zbet>0$, the sequence $(\grand)_{n\ge0} $
is relatively compact, and all its accumulation points belong to $\{I_{\zbet}=0\}$.}
\skip\\
\proof We proceed just as in the proof of the last proposition. For each $n\ge0$
we have
\bea
-\vni\log Z_{\nzbet,\per}&=&\vni\, I(\grand;Q_n)\\&&+\,\beta\,\grand(\Phi(R_n))-
\grand(\vni N_n)\log z\,.
\eea
The first term on the right-hand side is not less than $I(P^{(n)})$, where
$P^{(n)}$ is the invariant $\Lan$-average of the $\Lan$-periodic measure
$\hat P^{(n)}$ making the configurations in the blocks $\Lan+(2n+1)i,\,
i\in L$, independent with identical distribution $\grand$. Using Lemma 3.3
and the observation that $\grand(\rhoquadr(R_n))=\rhoquadr(\grand)<\infty$
we see further that the second term on the right-hand side is not less than
$\beta\,\Phi(\grand R_n)$. Combining this with (2.22) we obtain
$$
\limsup_{n\ti}\left[I(P^{(n)})+\beta\,\Phi(\grand R_n)-\rho(P^{(n)})
\log z \right]\le-p(\zbet)<\infty\,.
$$
The compactness of the level sets of $I$, the lower semicontinuity of $\Phi$,
and the asymptotic equivalence of $P^{(n)}, \grand R_n$, and $\grand$ thus
give the result. $\Box$
\skip
To conclude, we mention without proof that the entropy function $s(\,\cdot\,
,\,\cdot\,)$ is related to the pressure by the the classical Legendre
transformation as follows. For all $\zbet>0$,
\be
p(\zbet)=\max_{\nu\ge0,\eps\in\R}\left[s(\nu,\eps)-\beta\,\eps+
\nu\log z\right].
\ee
Conversely,
\be
s(\nu,\eps)=\min_{\beta\ge0,z>0}\left[p(\zbet)+\beta\,\eps-\nu\log z\right]
\ee
whenever $\nu>0$ and $s(\nu,\eps)>-\infty$.
\pagebreak
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\end{thebibliography}
\end{document}
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