\documentstyle[12pt,amssymb]{article}
%\documentstyle[10pt,amssymb]{book}
\pagestyle{myheadings}
\oddsidemargin0pt
\evensidemargin0pt
%\topmargin-.5cm
%\textheight18.5cm
%\textwidth12.5cm
\textheight21cm
\textwidth16cm
\parindent0.8cm
\frenchspacing
%\renewcommand{\baselinestretch}{1.2}
\renewcommand{\theequation}{{\arabic{section}.\arabic{equation}}}
\def\R{{\Bbb R}} %%
\def\N{{\Bbb N}} %%
\def\p{{\Bbb P}} %% Schoene Darstellungen der Zahlenmengen
\def\Z{{\Bbb Z}} %%
\def\Q{{\Bbb Q}} %%
\def\C{{\Bbb C}} %%
\def\E{{\Bbb E}} %%
\def\Va{{\Bbb V}} %%
\newcommand{\fcap}[2]{\refstepcounter{figure}
{\noindent \bf Figure #1:}{\hspace{.2cm}{\it #2}}}
\newcommand{\Remm}[1]{}
\newtheorem{theo}{Theorem}[section]
\newtheorem{lemma}[theo]{Lemma}
\newtheorem{cor}[theo]{Corollary}
\newtheorem{prop}[theo]{Proposition}
\newtheorem{defn}[theo]{Definition}
\newtheorem{research}[theo]{Open Problem}
\def\EndProof{{\begin{flushright}\vspace{-2mm}$\Box$\end{flushright}}}
\def\EndTheo{{\begin{flushright}\vspace{-2mm}$\blacksquare$\end{flushright}}}
%\numberwithin{equation}{section}
%\renewcommand{\thechapter}{\arabic{section}}
%\refstepcounter{equation}
%\makeindex
\begin{document}
{\bf
\title{ THERMODYNAMIC CHAOS AND THE STRUCTURE OF SHORT-RANGE SPIN GLASSES }
\author{ ~
\\ Charles M. Newman \thanks{Research supported in part
by NSF Grant DMS-9500868} \\ Courant
Institute of Mathematical Sciences \\
New York University \\ New York, NY 10012, USA \\ and
\\ Daniel L. Stein \thanks{Research supported in part
by DOE Grant DE-FG03-93ER25155} \\ Department of Physics \\
University of Arizona \\ Tucson, AZ 85721, USA}
\date{ October 1996 }}
%\address{}
\maketitle
\begin{abstract}
This paper presents an approach,
recently introduced by the authors and based on the
notion of ``metastates'', to the chaotic size
dependence expected in systems with many competing
pure states, and applies it to the Edwards-Anderson
(EA) spin glass model. We begin by reviewing the standard picture of
the EA model based on the Sherrington-Kirkpatrick (SK) model
and why that standard SK picture is untenable. We then
introduce metastates, which are the analogues of the
invariant probability measures describing chaotic dynamical
systems and discuss how they should appear in several
models simpler than the EA spin glass. Finally, we consider
possibilities for the nature of the EA metastate, including one
which is a nonstandard SK picture, and speculate on their
prospects. An appendix contains proofs used in
our construction of metastates and in the
earlier construction by Aizenman
and Wehr.
\end{abstract}
% {\em Keywords:}
%\pagenumbering{roman}
%\chapter*{Preface}
%\setcounter{page}{4}
%~
\newpage
%\setcounter{page}{5}
%\tableofcontents
%\pagenumbering{arabic}
\setcounter{section}{0}
%\setcounter{page}{1}
\section{Introduction}
\setcounter{equation}{0}
%\setcounter{page}{1}\pagenumbering{arabic}
%\markboth{CH. 1. INTRODUCTION}{CH. 1. INTRODUCTION}
The thermodynamic, or infinite volume, limit is a convenient mathematical
device for analyzing the thermal equilibrium properties of systems with a
finite, but very large, number of microscopic components. The
utility of this device has been particularly apparent in its
contribution to our conceptual understanding of phase transitions
in relatively simple models, such as the homogeneous Ising
ferromagnet (or equivalent lattice gas). In that context, it led
both to Onsager's exact calculation of the two-dimensional
free energy \cite{O1944} and to the Lee-Yang
mechanism (via partition function zeros) for obtaining
macroscopically distinct equations of state for different phases with a
single microscopic Hamiltonian \cite{YL1952}, \cite{LY1952}.
Behind these successes lie the related facts that
for such simple models, (a) one knows how to choose finite volume
boundary conditions (b.c.'s) so that the infinite volume
limit is a pure phase and (b) for b.c.'s not specially chosen to
yield a pure phase, one knows the resulting limiting mixture of pure phases.
For more complex models with an unknown relation between b.c.'s
and pure phases, the nature, meaning, and even utility of the infinite volume
limit (for local state observables, such as microscopic
correlations) may be more problematic. In particular, it was unclear how to
deal constructively with chaotic size dependence (CSD) \cite{NS1992},
the phenomenon that (with not specially chosen b.c.'s) there
may be many distinct limiting states along different subsequences of
volumes.
In this paper, we report on a new approach to the thermodynamic
limit \cite{NS1995a} which harnesses CSD by means of ``metastates'',
ensembles of (i.e., probability measures on) the possible
limiting (mixed) states. As in \cite{NS1995a}, we present this
approach primarily in the context of the Edwards-Anderson (EA)
spin glass model \cite{EA1975}, although our arguments apply to
a very wide class of disordered systems. In fact, the
metastate approach should be applicable also to nondisordered systems
with a complex structure of pure phases.
In addition to this introductory section, numbered 1, this paper
consists of seven more sections
and an appendix. In Section 2,
we introduce the EA model, the number $\cal N$ of pure phases for it,
and the main competing predictions from the physics literature
that when ${\cal N} \ne 1$, it must be two according to one view
and infinity according to another. The prediction that
${\cal N} = \infty$, based on the Sherrington-Kirkpatrick (SK) spin
glass model \cite{SK1975}, is part
of what we call an SK picture (of the
EA model). In Section 3, we present the ``standard'' SK picture
(which does not utilize the metastate approach) as in the
physics literature but made more precise, and then explain why it
breaks down \cite{NS1996b}. This breakdown leads us, in
Section 4, to our metastate approach. In Section 5, we digress to
discuss several examples of metastates in systems simpler than
the EA model and then in Section 6, we present several
possibilities concerning the nature of the metastate in the EA model
itself. One of these is a new possibility intermediate between
an SK picture and the prediction that ${\cal N} =2$.
To explore another of the possibilities, that a nonstandard SK picture,
within the framework of the metastate approach, might be valid
for the EA model, we begin by discussing, in Section 7, the use
of replicas and their overlaps in the metastate approach. In
Section 8, we discuss the nonstandard SK picture and its prospects.
Finally, the Appendix concerns various technical arguments
needed for the constructions of metastates in Section 4.
\section{The Edwards Anderson Model}
\setcounter{equation}{0}
%\setcounter{page}{1}\pagenumbering{arabic}
%\markboth{CH. 2. THE EDWARDS ANDERSON MODEL}{CH. 2. THE EDWARDS ANDERSON MODEL}
An Edwards-Anderson (EA) spin glass model
\cite{EA1975} is a disordered nearest neighbor Ising model on
$\Z^d$ whose couplings $J=(J_e:e\in \E^d)$ are
i.i.d. random variables (on some probability space
$(\Omega,{\cal F},\nu)$) with a common
symmetric distribution $\mu$ (i.e., $J_e$ and $-J_e$ are
equidistributed).
Here $\E^d$ denotes the set of nearest neighbor bonds $\{x,y\}$,
i.e., pairs of sites of $\Z^d$, whose
Euclidean distance $||x-y||$ equals $1$.
The most common examples are the Gaussian (where $\mu$
is a mean zero normal distribution) and the $\pm \check{J}$ (where
$\mu=\frac{1}{2}\delta_{\check{J}} +\frac{1}{2}\delta_{-\check{J}}$) models.
Unless otherwise noted, we place no
restrictions on $\mu$ beyond symmetry.
Let $\Lambda$ be a finite subset of $\Z^d$, for example, $\Lambda =
\Lambda_L= \{ -L, -L+1, \cdots ,L\}^d$. The EA model on
$\Lambda$ at inverse temperature $\beta \ge 0$
(with some b.c., temporarily denoted by *)
is a family $\{S_x:x\in \Lambda \}$ of random variables
(or spins) taking
values $+1$ or $-1$, whose joint distribution is
the finite volume Gibbs measure $P_{\Lambda,\beta}^*$
given by
\begin{equation}
P_{\Lambda,\beta}^*(\{s\})=Z_{\Lambda,\beta}^{*-1} \exp \{ -\beta
H_{\Lambda}^*(s)\}.
\end{equation}
Here $s\in {\cal S}_{\Lambda}=\{-1,+1\}^{\Lambda}$, $H_{\Lambda}^*(s)$ is
the finite
volume Hamiltonian, and the partition function, $Z_{\Lambda,\beta}^*$,
is such that $P_{\Lambda,\beta}^*({\cal
S}_{\Lambda})=1$.
The free b.c. Hamiltonian (with $* = f$) is
\begin{equation} \label{Hami}
H_{\Lambda}^f(s)= \sum_{\stackrel{x,y \in
\Lambda}{\{x,y\} \in \E^d}} -J_{\{x,y\}} s_x s_y.
\end{equation}
One type of b.c. (called a fixed b.c.) is the specification of a spin
configuration, $\bar{s}\in \{-1,+1\}^{\partial \Lambda}$, on the
boundary of $\Lambda$,
\begin{equation}
\partial \Lambda = \{ y \in \Z^d \setminus \Lambda: \{x,y\} \in \E^d
\mbox{ for some }x \in \Lambda \},
\end{equation}
and the replacement of (\ref{Hami}) by
\begin{equation}
H_{\Lambda}^{\bar{s}}(s)=H_{\Lambda}^{f}(s)
-\sum_{\stackrel{x \in \Lambda, y \in \partial
\Lambda}{\{x,y\} \in \E^d}} J_{\{x,y\}} s_x \bar{s}_y.
\end{equation}
A more general type of b.c. is
specified by a probability measure $\bar{\rho}$ on ${\cal S}_{\partial
\Lambda}$. The corresponding Gibbs distribution on ${\cal S}_{\Lambda}
$, denoted by $P_{\Lambda,\beta}^{\bar{\rho}}$, is the mixture of fixed
b.c. Gibbs distributions,
\begin{equation}\label{three1}
P_{\Lambda,\beta}^{\bar{\rho}}=\sum_{\bar{s} \in {\cal S}_{\partial
\Lambda}} \bar{\rho}(\{\bar{s}\})P_{\Lambda,\beta}^{\bar{s}}.
\end{equation}
Later on, we will introduce periodic b.c.'s when $\Lambda$
is the cube $\Lambda_L$. Meanwhile,
we recall that an infinite volume Gibbs state (at inverse temperature
$\beta$) for given $\{ J_e:e\in \E^d\}$ is any measure $P_{\beta}$ on
${\cal S}=\{-1,+1\}^{\Z^d}$ such that there is some sequence of b.c.'s
(on the cubes $\Lambda_L$), i.e., $\bar{\rho}_L$ on ${\cal S}_{\partial
\Lambda_L}$, such
that $P_{\beta}$ is the limit (in the sense of convergence of finite
dimensional distributions) of $P_{\Lambda_L,\beta}^{\bar{\rho}_L}$ as
$L\to \infty$. We remark that there is no loss of generality resulting
from the restriction to cubes or to this class of b.c.'s.
The intrinsic characterization of $P_{\beta}$ is that it satisfies the
Dobrushin-Lanford-Ruelle (DLR) equations (see, e.g., \cite{G1988} for a
more complete discussion and for historical references), i.e., for every
finite $\Lambda$, the conditional distribution of $P_{\beta}$,
conditioned on the $\sigma$-field generated by $\{s_x:x\in \Z^d\setminus
\Lambda \}$, is $P_{\Lambda,\beta}^{\bar{s}}$ where $\bar{s} \in {\cal
S}_{\partial \Lambda}$ is given by the (conditioned) values of $s_x$ for $x\in
\partial \Lambda$.
For a given $\omega$ and hence a given coupling configuration
$J=J(\omega)$, we may consider the set ${\cal G}={\cal
G}(J(\omega),\beta)$ of all infinite volume Gibbs states for $J(\omega)$
at inverse temperature $\beta$. In analyzing ${\cal G}$ it is natural to
consider the set of extremal (or pure) Gibbs states,
\begin{equation}\label{four3}
\mbox{ex }{\cal G} = {\cal G}\setminus \{\alpha P_1+(1-\alpha)P_2:~\alpha
\in (0,1);~P_1,P_2\in {\cal G}; ~P_1\neq P_2 \}.
\end{equation}
We define $N=N(J(\omega),\beta)$ to be the cardinality,
$|\mbox{ex }{\cal G}|$, of $\mbox{ex }{\cal G}$. $N$ can (a
priori) take any of the values $1,2,\dots,$ or $\infty$ (we do not
distinguish here between countably and uncountably infinite).
The next proposition shows that $N$ is ``self-averaged'', i.e.,
it is the same value for almost all $\omega$'s. The ``hard'' part
of the proof is to show measurability; self-averaging then
follows easily from invariance under spatial translations.
In the next section of the paper, the same translation
invariance argument will rule out the standard SK picture.
\begin{prop}\label{prop44}
$N(J(\omega),\beta)$ is measurable and a.s. equals a constant,
${\cal N}(d,\mu,\beta)$.
\end{prop}
{\it Proof.} We will use the fact from the general theory of Gibbs
distributions
(see Corollary 7.28 of \cite{G1988}) that $N\ge k$ if and only if
${\cal G}$ contains at least $k$ linearly independent measures
$P_1,\dots,P_k$. But this is so if and only if for some finite
$\Lambda\subset\Z^d$, there is linear independence of the
($2^{|\Lambda|}$-dimensional) vectors
$\vec{p}_\Lambda(P_1),\dots,\vec{p}_\Lambda(P_k)$, defined as
\begin{equation}\label{four4}
\vec{p}_\Lambda(P)=\left(P(s=t \mbox{ on }\Lambda):t\in {\cal S}_\Lambda
\right).
\end{equation}
Thus $N\ge k$ if and only if for some finite $\Lambda$, the closed
convex set,
\begin{equation}\label{four5}
G_\Lambda \equiv \left\{ \vec{p}_\Lambda(P):P\in {\cal G} \right\},
\end{equation}
is at least $k$-dimensional.
For $L$ sufficiently large (so that $\Lambda$ is contained in the cube
$\Lambda_L$), let us consider the finite volume approximation to
$G_\Lambda$, obtained by replacing ${\cal G}$ in (\ref{four5}) by the
finite volume Gibbs measures $P_{\Lambda_L,\beta}^{\bar{\rho}}$:
\begin{equation}\label{four6}
G_\Lambda^L\equiv \left\{
\vec{p}_\Lambda(P_{\Lambda_L,\beta}^{\bar{\rho}}): {\bar{\rho}}
\mbox{ is a probability measure on }{\cal S}_{\partial \Lambda_L} \right\}.
\end{equation}
Then $G_\Lambda^{L''}\subseteq G_\Lambda^{L'}$ for $L''>L'$ and $
G_\Lambda =\lim_{L\to \infty}G_\Lambda^L$. Let ${\cal R}_{\Lambda,k}$
denote the countable set of linearly independent $k$-tuples
$(\vec{p}_1,\dots \vec{p}_k)$ of vectors in $\R^{2^{|\Lambda|}}$ with
rational coordinates. Then $G_\Lambda$ is at least $k$-dimensional if
and only if there exists $(\vec{p}_1,\dots \vec{p}_k)\in {\cal
R}_{\Lambda,k}$ such that for each $j$, $\vec{p}_j\in G_\Lambda^L$ for
all (large) $L$.
Next we note that since $P_{\Lambda_L,\beta}^{\bar{\rho}}$ is just the convex
combination (corresponding to ${\bar{\rho}}$) of the fixed
b.c. $P_{\Lambda_L,\beta}^{\bar{s}}$'s,
\begin{equation}\label{four7}
G_\Lambda^L=\mbox{ convex hull
}\left(\vec{p}_\Lambda(P_{\Lambda_L,\beta}^{\bar{s}}): \bar{s} \in {\cal
S}_{\partial \Lambda_L} \right).
\end{equation}
From the definition of $P_{\Lambda_L,\beta}^{\bar{s}}$ we clearly have
that the countably many points
$\vec{p}_\Lambda(P_{\Lambda_L,\beta}^{\bar{s}})$, as $\Lambda$, $L$ and
$\bar{s}$ vary, are all measurable functions of $J$. Thus
\begin{eqnarray}\nonumber
\{ N \ge k \} &=& \big\{ \exists \Lambda,~ \exists (\vec{p}_1,\dots
\vec{p}_k)\in {\cal R}_{\Lambda,k},\\ \label{four8}
& & \quad \forall j, ~\forall \mbox{ large }L, \\ \nonumber
& & \quad \vec{p}_j \in \mbox{ convex hull
}\left(\vec{p}_\Lambda(P_{\Lambda_L,\beta}^{\bar{s}}): \bar{s} \in {\cal
S}_{\partial \Lambda_L} \right) \big\}
\end{eqnarray}
is measurable.
Finally, to complete the proof, note that the joint distribution
$\bar{\nu}$ of $J=(J_e:e\in \E^d)$ is an i.i.d. product measure which is
invariant and ergodic with respect to translations on $\Z^d$. Since
$N$ is a measurable function of $J$ which is clearly translation
invariant, it follows that $N$ is a.s. constant.
\EndProof
For a given $J$ and $\beta$, $N\ge 2$ if and only if there is
non-uniqueness of infinite volume Gibbs distributions. Thus
one may define a critical inverse temperature in terms of
the value of $N$. There are various ways to do this, one of
which is
\begin{eqnarray}\label{three58}
{\beta}_c^*=
\inf \{\beta \ge 0: {\cal N}\ge 2\}
=\inf \{ \beta \ge 0: \mbox{ a.s, there is }\\
\nonumber \mbox{ nonuniqueness of the infinite} \\
\nonumber \mbox{ volume Gibbs states for } \beta \}.\quad
\end{eqnarray}
If there is a.s. uniqueness of the infinite volume Gibbs state
for all finite $\beta$, then we set $\beta_c^* = \infty$.
Of course, this critical value is a function,
$\beta_c^*(d,\mu)$ of the spatial dimension and the common
distribution of the couplings.
It is not hard to see that for any $\mu$, $\beta_c^* = \infty$
for d=1. The critical value is also infinite for $d>1$ if
$\mu$ has too large of an atom at zero. It is also known that
for any $d$ and $\mu$, $\beta_c^*$ is strictly positive. (For
more details and for a discussion of some other definitions of
critical values, see Chapters 3 and 4 of \cite{N1997}.)
The general belief in the physics literature, based primarily on
numerical studies, is that there is some critical dimension
$d_c$ (probably with $d_c \le 3$ \cite{FS1990}) such that for $d\ge d_c$, the
critical $\beta$ is finite (at least for the standard examples of
$\mu$). It should be noted that in this literature, it is not usually
clear which of the
possible definitions of the critical value is under consideration.
It is also believed that, like in the random field
Ising model \cite{AW1990}, the critical $\beta$ is infinite for
$d=2$. Thus we have the following open problems.
\begin{research}\label{research42}
Determine
whether for all $\mu$, ${\beta}_c^*(2,\mu)=\infty$.
\end{research}
\begin{research}\label{research43} Determine whether for some $d$ and
some $\mu$,
%$\quad$
${\beta}_c^*(d,\mu)<\infty$.
\end{research}
Although it seems to be
generally believed that (at least for some $\mu$'s), ${\cal N}(d,\mu,\beta)\ge
2$ for $d\ge$ some $d_c$ and large $\beta$, there is a lively
controversy in the physics literature on the nature of the pure Gibbs
states and, in particular, on the value of ${\cal N}$ (when ${\cal N}\ge2$).
One side of the controversy is fairly easy to describe. This side
predicts, on the basis of nonrigorous scaling arguments \cite{M1984},
\cite{FH1986}, \cite{BM1987}, \cite{FH1988},
that ${\cal N}=2$ and that the two pure states, $P'$ and $P''$, are global flips
of each other, i.e., that the mapping $s\to -s$ on ${\cal S}$ transforms
$P'$ into $P''$ and vice versa. (See \cite{BF1986} for different
arguments that predict ${\cal N}=2$ only for $d=3$ and
\cite{E1990} for a critique of the scaling argument prediction
that ${\cal N}=2$ for all $d$.)
This would be analogous to the situation
for the homogeneous $d=2$ Ising ferromagnet \cite{A1980},
\cite{H1981}. The local magnetizations $\langle S_x \rangle'$ (where
$\langle \cdot \rangle'$ denotes expectation w.r.t. $P'$) and $\langle
S_x \rangle''=- \langle S_x \rangle'$
would depend on both $x$ and $\omega$ and would be nonvanishing (at
least for some $x$'s with a positive density in $\Z^d$). The mean
magnetization should vanish; i.e., w.r.t. $P'$,
\begin{equation}\label{four9}
\lim_{L\to\infty} |\Lambda_L|^{-1} \sum_{y\in
\Lambda_L}S_y=\lim_{L\to\infty}|\Lambda_L|^{-1} \sum_{y\in \Lambda_L}
\langle S_y \rangle'=0.
\end{equation}
However, the EA order parameter should be a strictly
positive constant, $r_{EA}=r_{EA}(d,\mu,\beta)$, given by
\begin{equation}\label{four10}
r_{EA}=\lim_{L\to\infty} |\Lambda_L|^{-1} \sum_{y\in
\Lambda_L}\left[\langle S_y \rangle'\right]^2=E\left( [\langle S_x
\rangle' ]^2\right),
\end{equation}
where $E$ denotes expectation with respect to $\nu$ (on the probability
space of $J$).
One feature of the other side of the controversy is also easy to
describe $-$ i.e., the prediction that ${\cal N}=\infty$. Other features
are more difficult and much of this paper will be devoted to
presenting and analyzing different interpretations of how ${\cal N}=\infty$
could or should manifest itself. Before we begin that, we
list an obvious open problem.
\begin{research}\label{research45}
Determine whether ${\cal N}(d,\mu,\beta)=\infty$, or at least whether
${\cal N}(d,\mu,\beta)>2$, for some $d$, $\mu$ and $\beta$.
\end{research}
The prediction that ${\cal N}=\infty$ is based firstly on the notion that a
short-ranged spin glass like the EA model should behave qualitatively
like an ``infinite-ranged'' or ``mean-field'' model and based secondly
on the nonrigorous analysis by Parisi and others \cite{P1979},
\cite{P1983}, \cite{HJY1983}, \cite{MPSTV1984} of the
Sherrington-Kirkpatrick (SK) model \cite{SK1975}, the standard mean-field
spin glass. This line of reasoning leads not only to the prediction that
${\cal N}=\infty$ (for appropriate $d$, $\mu$ and $\beta$) but also to many
other predictions concerning the Gibbs states for large finite volume
and for infinite volume.
These predictions, taken as a whole, constitute what we call an SK
picture of the EA model. Unfortunately the predictions (other than that
${\cal N}=\infty$) have not been formulated in the literature with much
precision so that before attacking the problem of whether the SK picture
(of the EA model) is valid, it is first necessary to consider the
question, ``what is the SK picture?''. One of our main objectives here,
following \cite{NS1992}, \cite{NS1996b}, \cite{NS1995a}, is to answer
that question. To begin, we start the next
section with a brief introduction to the SK
model itself and to Parisi's analysis of it. For a comprehensive
discussion, the reader is referred to \cite{MPV1987}.
\section{The Standard SK Picture}
\setcounter{equation}{0}
%\markboth{CH. 3. THESTANDARD SK PICTURE}{CH. 3. THE STANDARD SK PICTURE}
Unlike the EA model, the SK model has
no a priori connection with the lattice $\Z^d$ and so it is conventional
to replace subsets of $\Z^d$ by subsets of $\N$,
the set of positive integers. The SK
model of size $n$ at inverse temperature $\beta$ has the Gibbs measure
on ${\cal S}_n=\{-1,+1\}^n$,
\begin{equation}\label{four11}
P_{n,\beta}(\{s\})=Z_{n,\beta}^{-1} \exp \left( \beta \sum_{i=1}^{n-1}
\sum_{j=i+1}^{n} J_{ij}^n s_i s_j \right),
\end{equation}
with the $n$-dependent couplings given by
\begin{equation}\label{four12}
J_{ij}^n= n^{-1/2}K_{ij}\quad \mbox{ for }1\le i \beta_c^{CW}$, the limit is
$\frac{1}{2}\delta_{M^\ast}+\frac{1}{2}\delta_{-M^\ast}$, where $M^\ast
=M^\ast(\beta)$, the spontaneous magnetization, is strictly positive for
$\beta> \beta_c^{CW}$. In the SK model, the magnetization per site is
not a fruitful choice of variable since, due to sign cancellations, it
should have a trivial limiting distribution, $\delta_0$, for all
$\beta$. Indeed, Parisi's analysis of the free energy for the SK model
led him to study the distribution of a different random variable, the
replica overlap.
The replica overlap is the random variable (on ${\cal S}_n\times{\cal
S}_n=\{(s^1,s^2)\}$ with probability measure $P_{n,\beta}\times
P_{n,\beta}$)
\begin{equation}\label{four15}
R_n=\frac{1}{n} \sum_{i=1}^n s_i^1 s_i^2.
\end{equation}
The formal density of its distribution (on the interval $[-1,1]$)
is denoted ${\cal P}_n(r)$. For $\beta$ below a critical value
$\beta_c^{SK}$, the $n\to \infty$ limit of ${\cal P}_n(r)$ is
$\delta(r)$, the trivial point density at $0$. But above $\beta_c^{SK}$,
Parisi found highly nontrivial behavior: namely, that as $n\to \infty$,
${\cal P}_n$ approximates a sum of many delta functions, at locations
and with weights which depend on $\omega$, with the weights {\it not}
tending to zero and with the dependence of the locations and weights on
$\omega$ also {\it not} tending to zero as $n\to \infty$.
The usual explanation given for this behavior (see, e.g.,
\cite{MPV1987}) is that, as $n\to \infty$, the Gibbs measure
$P_{n,\beta}$ has a decomposition into many pure states
$P^\alpha=P_\beta^\alpha$,
\begin{equation}\label{four16}
P_{n,\beta}\approx \sum_\alpha W^\alpha P^\alpha,
\end{equation}
with weights $W^\alpha$ depending on $\omega$, with neither they nor
their dependence on $\omega$ tending to zero.
[Warning: The reader should not be alarmed if the meaning of a pure
state in the SK context is not precisely clear; it is certainly not
clear to the authors.]
If the replicas $s^1$ and $s^2$ were chosen from $P^\alpha \times
P^\gamma$ rather than from $P_{n,\beta}\times P_{n,\beta}$, then, as
$n\to \infty$, the overlap would have a point density
\begin{equation}\label{four17}
\delta(r-r_{\alpha\gamma}), \quad \quad r_{\alpha\gamma}\approx
\frac{1}{n} \sum_{i=1}^n \langle S_i \rangle^\alpha \langle S_i
\rangle^\gamma,
\end{equation}
and so
\begin{equation}\label{four18}
{\cal P}_n(r)\approx \sum_\alpha \sum_\gamma W^\alpha W^\gamma
\delta(r-r_{\alpha\gamma}).
\end{equation}
Like the $W^\alpha$'s, the $r_{\alpha\gamma}$'s depend on $\omega$ with a
dependence not tending to zero. This persistence in the dependence of
various quantities on $\omega$ in the infinite volume
limit, $n\to\infty$, is called non-self-averaging (NSA) and is one of
the essential features of Parisi's analysis.
(For a rigorous approach to NSA in the SK model, see
\cite{PS1991}.)
A second essential feature is that there are many $\alpha$'s (with
non-negligible weights) appearing in (\ref{four16}) so that the two
replicas have non-negligible probability of appearing in different (and
``unrelated'') pure states $\alpha$ and $\gamma$ (with overlap value
$r_{\alpha\gamma}$). This is called replica symmetry breaking (RSB). By
unrelated, we mean that $\gamma$ is neither $\alpha$ nor the ``negative''
of $\alpha$, i.e., that $P^\gamma$ is not $P^\alpha$ and is also not the
global spin flip of $P^\alpha$. When $\gamma=\alpha$, $r_{\alpha\alpha}$
should, for all $\alpha$, be the Edwards-Anderson order parameter
$r_{EA}$ (the analogue of (\ref{four10}), but for the SK model), while
for $\gamma=-\alpha$, $r_{\alpha\gamma}=-r_{EA}$. The value of $r_{EA}$
(unlike other $r_{\alpha\gamma}$'s) should {\it not} depend on $\omega$.
A third essential feature, closely related to NSA, is that the discrete
nature of the Parisi order parameter distribution as a countable sum of
delta functions is only for fixed $\omega$. As $\omega$ varies, the
$r_{\alpha\gamma}$'s vary in such a way that, as $n\to\infty$, $\bar{\cal
P}_n(r)$, the average over the couplings (i.e., over the underlying
probability measure $\nu$ on $\Omega$) is continuous (except for two
delta functions at $\pm r_{EA}$ since those locations do not depend on
$\omega$).
A fourth essential feature, closely related to, but going well beyond,
the discreteness of the order parameter distribution, is ultrametricity
of the $r_{\alpha\gamma}$'s (for fixed $\omega$). Here one regards
$d_{\alpha\gamma}\equiv r_{EA} -r_{\alpha\gamma}$ as defining a metric on
the pure states. The ultrametric property of $d_{\alpha\gamma}$ is that
among any three pure states $\alpha$, $\gamma$ and $\delta$, the largest
two of $d_{\alpha\gamma}$, $d_{\alpha\delta}$, $d_{\gamma\delta}$ are
equal.
In formulating a precise interpretation of the SK picture of the EA
model, it is clear that a primary role will be played by an EA model
analogue of the approximate pure state decomposition (\ref{four16}). In our EA
analogues of (\ref{four16}), we will replace $P_{n,\beta}$ on the LHS by
$P_L=P_{\Lambda_L,\beta}^{per}$, the ($\omega$-dependent) Gibbs measure
for the EA model on the cube $\Lambda_L$ with periodic b.c.'s (although
for many purposes, free b.c.'s or, indeed, any b.c. not depending on
$\omega$ could equally well be chosen). The EA model has a great advantage
over the SK model in that there is already a well defined meaning of
pure states as extremal infinite volume Gibbs states. Thus in the EA
model, the role of pure states will be played by $\dots$ the pure
states. The real issue for EA analogues of (\ref{four16}) is
interpreting the approximate equality.
The most straightforward interpretation of the SK picture, which we call
the standard SK picture, is also the one that most closely matches the
presentations in the
physics literature. In this picture, (\ref{four16}) is replaced by the
identity
\begin{equation}\label{four19}
P_{\cal J}=\sum_\alpha W^\alpha_{\cal J} P^\alpha_{\cal J},
\end{equation}
where ${\cal J}=({\cal J}_e:e\in \E^d)$ represents a particular
configuration of the random couplings $J=(J_e:e\in \E^d)$, $P_{\cal J}$
is an infinite volume Gibbs state for ${\cal J}$ (and some fixed
$\beta$) obtained in some natural way from the finite volume periodic
b.c. Gibbs states $P_L=P_{{\cal J},L}$ by letting $L\to \infty$, and the
$P^\alpha_{\cal J}$'s are pure (i.e., extremal infinite volume) Gibbs states
for that same ${\cal J}$. The identity (\ref{four19}) is to be valid for
$\bar{\nu}$-a.e. ${\cal J}$, where $\bar{\nu}$ is the joint distribution
of the $J_e$'s. The replica overlap in this picture is the random
variable (on ${\cal S}\times{\cal S}$ with probability measure $P_{\cal
J}\times P_{\cal J}$),
\begin{equation}\label{four20}
R_{\cal J}=\lim_{L\to\infty}|\Lambda_L|^{-1}\sum_{x\in
\Lambda_L}s_x^1s_x^2,
\end{equation}
and (the formal density of) its distribution is the Parisi overlap
distribution ${\cal P}_{\cal J}$. The limit over cubes in (\ref{four20}) is to
exist, a.s. with respect to $P_{\cal J}\times P_{\cal J}$, for
$\bar{\nu}$-a.e. ${\cal J}$ and the overlap distribution ${\cal P}_{\cal J}$ is
to depend measurably on ${\cal J}$.
If all this were so, then it would follow from (\ref{four19}) and the
tail $\sigma$-field triviality of pure Gibbs states (see Chapter 7 of
\cite{G1988}), that
\begin{equation}\label{four21}
{\cal P}_{\cal J}(r)=\sum_\alpha \sum_\gamma W_{\cal J}^\alpha W_{\cal
J}^\gamma \delta (r-r^{\cal J}_{\alpha \gamma}),
\end{equation}
with
\begin{equation}\label{four22}
r^{\cal J}_{\alpha \gamma}=\lim_{L\to\infty}|\Lambda_L|^{-1}
\sum_{x\in\Lambda_L} \langle S_x \rangle_{\cal J}^\alpha \langle S_x
\rangle_{\cal J}^\gamma.
\end{equation}
The standard SK picture thus predicts that the overlap distribution
${\cal P}_{\cal J}$, obtained from $P_{\cal J}$ in this way, has the
following four essential features. (1) NSA $-$ ${\cal P}_{\cal J}$ does depend
on ${\cal J}$; (2) nontrivial discreteness $-$ ${\cal P}_{\cal J}$ is a sum of
(countably) infinitely many delta functions; (3) continuity of its ${\cal
J}$-average $-$ $\bar{{\cal P}}\equiv \int {\cal P}_{\cal J}
d\bar{\nu}({\cal J})$ is continuous (except for two $\delta$-functions
at $\pm r_{EA}$, whose weights add up to less than one); (4)
ultrametricity of the $r^{\cal J}_{\alpha \gamma}$'s.
We remark that although the discreteness of ${\cal P}_{\cal J}$
is essential to this SK picture, it has been pointed out
to us by A.~van~Enter that this could be
the case without discreteness of the pure state decomposition
(\ref{four19}); indeed, such a situation occurs in some
deterministic models considered in \cite{EnHM1992}.
However, there is another feature of the SK picture,
as usually presented in the physics literature,
which suggests discreteness of the pure state decomposition.
This is generally described in terms of there being free
energy gaps of order unity between the low-lying states
in any (large) volume,
accompanied by an exponentially increasing density
of states as the free energy increases from the
bottom of the spectrum \cite{MPV1985, DT1985, B1992}.
Throughout this paper, we will take a countable pure state
decomposition in various SK pictures. However, the reader
should bear in mind that, at least in principle,
it might be possible to have a continuum pure state decomposition
without violating those features of the
SK pictures that concern
only the overlaps.
Now that we have formulated the standard SK picture, we can ask whether
this picture of the EA model can be valid (for some dimensions and some
temperatures). This question has two parts. First, does there exist some
natural construction which begins with the finite volume states,
$P_{{\cal J},L}$, takes $L\to \infty$, and ends with an infinite volume
state, $P_{\cal J}$, and its accompanying overlap distribution ${\cal
P}_{\cal J}$? Second, can such a ${\cal P}_{\cal J}$ exhibit all the
essential features of the SK picture? The answers to these two parts,
given in \cite{NS1996b} are, respectively, yes and no, as we now
explain. We will not formalize the answer to the first part as a
proposition or theorem because it will be implicitly included
as a part
of the more comprehensive ``metastate'' approach given
in the next section.
We begin our answer to the first part of the question by noting that we
cannot simply
fix ${\cal J}$ and take an ordinary limit of the finite cube, periodic
b.c. state $P_{{\cal J},L}$, as $L\to \infty$. Unlike, say,
the $d=2$
homogeneous Ising ferromagnet, where such a limit exists (and equals
$\frac{1}{2} P^++\frac{1}{2} P^-$) by spin flip symmetry considerations
(and the fact that $P^+$ and $P^-$ are the only pure states
\cite{A1980}, \cite{H1981}), there is no guarantee for a spin glass that
there is a well defined limit. By compactness arguments, one can easily
obtain, for each ${\cal J}$, convergence along subsequences of
$L$'s. But these subsequences may be ${\cal J}$-dependent and there
seems to be no natural way to patch together the limits for different
${\cal J}$'s to yield $P_{\cal J}$. This is more than a technical
problem. As first discussed in \cite{NS1992} and as we will explain
below, when there are many competing pure states (as in an SK picture),
there should be chaotic size dependence (CSD) $-$ i.e., the existence,
for typical configurations of ${\cal J}$, of different limits along
different (${\cal J}$-dependent) subsequences.
In spite of the ``problem'' of CSD, a limit can be taken, not by fixing
${\cal J}$, but by considering the joint distribution $\bar{\nu}({\cal
J}) \times P_{{\cal J},L}((s_x:x\in \Lambda_L))$, of the $J_e$'s and the
$S_x$'s. (We note that such joint distribution limits were
considered, implicitly or explicitly, in \cite{Le1977},
\cite{Co1989}, \cite{GKN1992} and \cite{S1995}.)
That is, by choosing a subsequence of $L$'s (not depending on
${\cal J}$), one has convergence of all the finite dimensional
distributions of the $J_e$'s and $S_x$'s to those of a probability measure
on $\R^{\E^d}\times {\cal S}$, whose marginal distribution of ${\cal J}$
is $\bar{\nu}$ and whose conditional distribution of $s$, given ${\cal
J}$, is some $P_{\cal J}$. All this follows from standard compactness
arguments with $P_{\cal J}$ a probability measure on ${\cal S}$ defined
for $\bar{\nu}$-a.e. ${\cal J}$ and depending measurably on ${\cal
J}$. What doesn't follow from general compactness arguments is that (for
$\bar{\nu}$-a.e. ${\cal J}$) $P_{\cal J}$ is an infinite volume Gibbs
state for ${\cal J}$. This however can be shown either directly,
or as a corollary of a more comprehensive result from \cite{AW1990} that
is also presented below (see Theorem \ref{prop47} and the
remark following it).
This construction has certain translation invariance properties that are
important, both technically and conceptually. Because of the periodic
b.c.'s on the cube $\Lambda_L$, the couplings and spins are really
defined on a (discrete) torus of size $L$, with a joint distribution
invariant under torus translations. This implies that any
(subsequence) limit joint distribution on $\R^{\E^d}\times{\cal S}$ is
invariant under translations of $\Z^d$, which in turn implies that
$P_{\cal J}$ is translation covariant; i.e., under the translation of
${\cal J}$ to ${\cal J}^a$, where ${\cal J}^a_{\{x,y\}}={\cal
J}_{\{x+a,y+a\}}$, $P_{\cal J}$ transforms so that
\begin{equation}\label{four23}
P_{{\cal J}^a}(S_{x_1}=s_1,\dots,S_{x_m}=s_m)=P_{\cal
J}(S_{x_1-a}=s_1,\dots S_{x_m-a}=s_m).
\end{equation}
The conceptual significance of translation covariance is that the mapping
from ${\cal J}$ to $P_{\cal J}$, being a natural one, should not (and in
this construction does not) depend on the choice of an origin. The
technical significance is that it implies that the joint measure for ${\cal
J}$ and the two replicas $s^1$ and $s^2$, $\bar{\nu}({\cal J})P_{\cal
J}(s^1)P_{\cal J}(s^2)$, on $\R^{\E^d}\times{\cal S}\times{\cal S}$ is
translation invariant (under $({\cal J},S^1,S^2)\to({\cal
J}^a,S^{1a},S^{2a})$) which implies, by the ergodic theorem, that
$|\Lambda_L|^{-1} \sum_{x\in\Lambda_L}s_x^1s_x^2$ has an a.s. limit $R$
(with respect to $\bar{\nu}({\cal J})P_{\cal
J}(s^1)P_{\cal J}(s^2)$) and thus the $R_{\cal J}$ of (\ref{four20})
exists a.s. (with respect to $P_{\cal J}\times P_{\cal J}$) for
$\bar{\nu}$-a.e. ${\cal J}$, as desired. ${\cal P}_{\cal J}$, the
distribution of $R_{\cal J}$, then exists for $\bar{\nu}$-a.e. ${\cal
J}$ and depends measurably on ${\cal J}$. Indeed ${\cal P}_{\cal J}$ is
simply the conditional distribution of the random variable $R$, given
${\cal J}$.
We have answered the first part of our question on the validity of the
standard SK picture by showing that, yes, there does exist a natural
$P_{\cal J}$ and ${\cal P}_{\cal J}$, that are related as required by
that picture and that depend on ${\cal J}$ measurably. To begin our
answer to the second part of the question, we see what the translation
covariance of $P_{\cal J}$ implies about ${\cal P}_{\cal J}$. By
translation covariance, $R_{{\cal J}^a}$ is equidistributed with the random
variable (on ${\cal S}\times{\cal S}$ with measure $P_{\cal
J}\times P_{\cal J}$)
\begin{equation}\label{four24}
R^{-a}_{\cal J}\equiv \lim_{L\to\infty}|\Lambda_L|^{-1}\sum_{x\in
\Lambda_L} s_{x-a}^1s_{x-a}^2=R_{\cal J}.
\end{equation}
Thus ${\cal P}_{{\cal J}^a}={\cal P}_{\cal J}$ for
$\bar{\nu}$-a.e. ${\cal J}$ and all $a\in Z^d$; i.e., ${\cal P}_{\cal
J}$ is translation
invariant. As in the case of the translation covariance of $P_{\cal J}$,
the translation invariance of ${\cal P}_{\cal J}$ has the conceptual
significance that a natural object like the Parisi order parameter
distribution should not (and in this construction does not) depend on
the choice of an origin. But it also
has an important technical significance which, in the next proposition,
explains why the answer to the second part of our question on the
validity of the standard SK picture is, no, such a ${\cal P}_{\cal J}$
{\it cannot} exhibit all the essential features of the SK picture.
\begin{prop}[\cite{NS1996b}]\label{prop45} If ${\cal P}_{\cal J}$ is
translation invariant, then it is self-\linebreak averaged; i.e., it
equals a fixed
probability measure ${\cal P}$ on $[-1,1]$ for $\bar{\nu}$-a.e. ${\cal
J}$. Thus it does not exhibit (1) non-self-averaging and consequently
also does not exhibit at least one of (2) nontrivial discreteness or (3)
continuity of its ${\cal J}$-average.
\end{prop}
{\it Proof.} Consider for any $k$, the moment $\int_{-1}^1 q^k {\cal
P}_{\cal J}(q)dq$. This is a measurable function of ${\cal J}$ defined
for $\bar{\nu}$-a.e. ${\cal J}$ which is invariant under the translation
${\cal J}\to {\cal J}^a$ for every $a\in \Z^d$. By the spatial
translation invariance and ergodicity of $\bar{\nu}$, this implies that
this function is $\bar{\nu}$-a.s. a constant. These moments, for all
$k$, determine ${\cal P}_{\cal J}$; thus ${\cal P}_{\cal J}$ itself is
$\bar{\nu}$-a.s. a constant ${\cal P}$. Since ${\cal P}=\int {\cal
P}_{\cal J} d\bar{\nu}({\cal J})$, the rest of the proposition follows.
\EndProof
\section{Chaotic Size Dependence and Metastates}
%\markboth{CH. 4. CSD AND METASTATES}{CH. 4. CSD AND METASTATES}
\setcounter{equation}{0}
Technically, the feature of the standard SK picture which led to its
demise was the translation covariance of the infinite volume state
$P_{\cal J}$. In pursuing other interpretations of the SK picture, and
in analyzing disordered systems more generally, we will not give
up translation covariance, but we will give up the idea that a
disordered system, in the infinite volume limit, should necessarily be
described by a single $P_{\cal J}$, i.e., by a function from coupling
configurations
to {\it single} infinite volume Gibbs states. Indeed, rather than
finessing CSD (by constructing our $P_{\cal J}$), we will try to
understand (or at least describe) CSD by analyzing the way in which
$P_{{\cal J},L}$ samples from its various possible limits as
$L\to\infty$. A major contribution of \cite{NS1995a} was the proposal
that this sampling is naturally understood in terms of a ``metastate'',
a probability measure $\kappa_{\cal J}$ on the infinite volume Gibbs
states for the given ${\cal J}$. (We will give a more precise definition
below.)
This proposal of \cite{NS1995a} was based on an analogy with chaotic
deterministic dynamical systems, where the chaotic motion along a
deterministic orbit is analyzed in terms of some appropriately selected
probability measure, invariant under the dynamics. Time along the orbit
is replaced, in our context, by $L$ and the state space (or
configuration space or phase space) of the dynamical system is replaced
by the space of Gibbs states (for a fixed ${\cal J}$). We will
delay until Section 8 any discussion of
the issue of what, in disordered systems, replaces
invariance of the probability measure under the dynamics.
Rather we will now
explain how the same metastate can be constructed by two different
approaches, one based on the randomness of the ${\cal J}$'s and the
other based on CSD for a fixed ${\cal J}$.
The approach based on ${\cal J}$-randomness is due to Aizenman and Wehr
\linebreak
\cite{AW1990}. This approach is analogous to the construction of
$P_{\cal J}$ described in the last section,
except that instead of considering the
random pair $(J,S)$, distributed for finite $L$ by $\bar{\nu}({\cal J})\times
P_{{\cal J},L}$, and then taking the limit (along a subsequence) of
finite dimensional distributions, one considers the random pair
$(J,P_{J,L})$, defined on the underlying probability space
$(\Omega,{\cal F},\nu)$ of $J$, and takes the limit of its finite
dimensional distributions. The finite dimensional distributions can be
defined in a number of equivalent ways. We will consider (for $P_{J,L}$)
the random (because of $J$) probabilities of cylinder
sets. I.e., for each finite $\Lambda\subset\Z^d$ and $s\in {\cal
S}_\Lambda=\{-1,+1\}^\Lambda$ (we denote by ${\cal A}$ the set of all such
pairs $(\Lambda,s)$) and for $L$ sufficiently large so that
$\Lambda\subseteq\Lambda_L$, we define the random variable (on
$(\Omega,{\cal F},\nu)$)
\begin{equation}\label{four25}
Q_{(\Lambda,s)}^{(L)}=P_{J,L}\left( \left\{ s'\in {\cal
S}_{\Lambda_L}:s_x'=s_x~\forall x \in \Lambda \right\}\right).
\end{equation}
Let $\kappa^\dagger$ denote a probability measure on $\Omega^\dagger =
\R^{\E^d}\times\R^{\cal A}$, with the product Borel $\sigma$-field
${\cal F}^\dagger$. We say that $(J,P_{J,L})\to \kappa^\dagger$ as $L\to
\infty$, if each of the finite dimensional distributions of $\left(
J_e,Q^{(L)}_{(\Lambda,s)}: e\in \E^d,~ (\Lambda,s)\in{\cal A}\right)$
converges as $L\to\infty$ to the corresponding finite dimensional
marginal distribution of $\kappa^\dagger$.
\begin{research} \label{research46} Prove that $(J,P_{J,L})\to
\kappa^\dagger$ for some probability measure $\kappa^\dagger$ on
$(\Omega^\dagger, {\cal F}^\dagger)$.
\end{research}
Although convergence of $(J,P_{J,L})$ has not been proved, it can be
shown \cite{AW1990} (see also Lemmas \ref{lemmaB2}, \ref{lemmaB3} and
\ref{lemmaB4} of the Appendix) that there is sequential compactness and
that every subsequence limit $\kappa^\dagger$ has a conditional
distribution $\kappa_{\cal J}$ (of $q\in \Omega^1\equiv \R^{\cal A}$,
given ${\cal J}\in \Omega^0\equiv \R^{\E^d}$) that, for
$\bar{\nu}$-a.e. ${\cal J}$, is supported on infinite volume Gibbs
distributions for that ${\cal J}$. Thus we have the following result.
\begin{theo}[\cite{AW1990}] \label{prop47} There exists a subsequence
$L_n$ of the $L$'s such that
$(J,P_{J,L_n})$ $ \to \kappa^\dagger$. Here
$\kappa^\dagger$ is a probability measure on $(\Omega^\dagger,{\cal
F}^\dagger)$ whose marginal distribution $\kappa_{\cal J}$ (of $q$,
given ${\cal J}$) satisfies:
%\begin{equation}
%\label{four26}
$\mbox{ for }\bar{\nu}\mbox{-a.e. }{\cal J},$
$\kappa_{\cal J}\left(\{q:q\mbox{ is an infinite volume Gibbs
state for }{\cal J} \} \right)=1.$
%\end{equation}
\end{theo}
{\it Proof.} This follows directly from Lemmas \ref{lemmaB2},
\ref{lemmaB3} and \ref{lemmaB4} of the Appendix.
\EndProof
\setcounter{theo}{3}
\noindent
{\bf Remark 4.3} There is more than an analogy relating the
construction of $\kappa_{\cal J}$ given in the last proposition to the
construction of the Gibbs distribution $P_{\cal J}$ given earlier. By
restricting the function $f$ appearing in (\ref{B8}) of the Appendix to be
linear in $q$, we see that, as $n\to\infty$,
\begin{equation}\label{four27}
E\big(g(J)\langle S_A \rangle_{J,L_n} \big) \to \int_{\Omega^0}
g({\cal J}) \left[ \int_{\Omega^1}\langle S_A\rangle_q d\kappa_{\cal
J}(q)\right] d\bar{\nu}({\cal J}),
\end{equation}
where $g$ is a (continuous, bounded) function of finitely many couplings,
$A$ is a finite subset of $\Z^d$, $S_A=\prod_{x\in A}S_x$, $\langle \cdot
\rangle_{J,L_n}$ denotes expectation with respect to $P_{J,L_n}$, and
$\langle \cdot \rangle_q$ denotes the expectation with respect to $q$
(for $q$ a probability measure on ${\cal S}$). On the other hand, the
construction of $P_{\cal J}$ yields
\begin{equation}\label{four28}
E\big(g(J)\langle S_A \rangle_{J,L_n} \big) \to \int_{\Omega^0}
g({\cal J}) \langle S_A \rangle_{P_{\cal J}} d\bar{\nu}({\cal J}).
\end{equation}
Since these last two equations are valid for general $g$ and $A$, we
conclude (see \cite{AW1990}) that the state $P_{\cal J}$ is the mean of
the metastate $\kappa_{\cal J}$, i.e.,
\begin{equation}\label{four}
P_{\cal J}=\int_{\Omega^1} q~d\kappa_{\cal J}(q).
\end{equation}
Theorem~\ref{prop47} then implies that $P_{\cal J}$ is indeed a Gibbs
distribution.
The second approach to constructing a metastate takes a fixed ${\cal J}$
and replaces ${\cal J}$-randomness, roughly speaking, by regarding $L$
as random, i.e., by considering the empirical distribution of $P_{J,L}$
as $L$ varies. We define the empirical distribution along a given
subsequence $L_n$
of the $L$'s as follows. First, we make the convention that
$Q^{(L)}_{(\Lambda, s)}$, defined by (\ref{four25}) for $\Lambda
\subseteq \Lambda_L$, is zero when $\Lambda \not\subseteq \Lambda_L$,
and then we define $\vec{Q}^{(L)}$ as the $\Omega^1$-valued random
variable (on the underlying probability space $(\Omega,{\cal F},\nu)$ of
$J$),
\begin{equation}\label{four30}
\vec{Q}^{(L)}=\left( Q^{(L)}_{(\Lambda, s)}:(\Lambda,s)\in {\cal
A}\right).
\end{equation}
The empirical distribution, along the subsequence $(L_n)$, is the random
discrete measure on $\Omega^1$,
\begin{equation}\label{four31}
\kappa_J^K\left( (L_n) \right) =\frac{1}{K}\sum_{k=1}^K
\delta_{\vec{Q}^{(L_k)}} .
\end{equation}
As in the first approach to metastates, the empirical distribution
approach has an open problem about convergence (which
will be discussed again in later sections of the paper)
along with a partial
solution based on taking subsequences. For probability measures
$\kappa_m'$ and $\kappa'$ on $\Omega^1$, we say
$\kappa_m'\to\kappa'$ if each of the finite
dimensional marginal distributions of
$\kappa_m'$ converges as $m\to \infty$ to
the corresponding marginal distribution of
$\kappa'$.
\begin{research}\label{research49} Determine whether for $\nu$-a.e. $\omega$,
$\kappa_{J(\omega)}^K((L_n\equiv n))\to\kappa_{\omega}'$, as
$K\to\infty$, for some probability measure $\kappa_{\omega}'$ on
$\Omega^1$.
\end{research}
The partial solution to this problem is based on Theorem
\ref{prop47}. If $(L_n)$ is a subsequence such that $(J,P_{J,L_n})\to
\kappa^\dagger$, it can be shown (see Proposition \ref{propB6} of
the Appendix) that by taking a subsequence $n_k$ of the $n$'s and a
subsequence $K_m$ of the $K$'s, that for $\nu$-a.e. $\omega$,
$\kappa_{J(\omega)}^{K_m}((L_{n_k}))$ converges as $m\to\infty$ to
$\kappa_{J(\omega)}$, where $\kappa_{\cal J}$ is the conditional
distribution of $\kappa^\dagger$ (of $q$, given ${\cal J}$). Combining
this with Theorem \ref{prop47}, we have the following result, announced in
\cite{NS1995a}.
\begin{theo}[\cite{NS1995a}]\label{theo410} There is a sub-subsequence
$L_{n_k}$ of the $L_n$'s of Theorem \ref{prop47} such that along
some subsequence $K_m$ of the $K$'s,
\begin{equation}\label{four32}
\mbox{ for $\nu$-a.e. }\omega, \quad \kappa_{J(\omega)}^{K_m}((L_{n_k}))
\to \kappa_{J(\omega)} \quad \mbox{ as } m \to \infty,
\end{equation}
where $\kappa_{\cal J}$ is the same metastate as in Theorem
\ref{prop47}.
\end{theo}
{\it Proof.} This result follows from Proposition \ref{propB6} of the
Appendix.
\EndProof
\section{Metastates in Simpler Models}
%\markboth{CH. 5. METASTATES IN SIMPLER MODELS}{CH. 5. METASTATES IN SIMPLER MODELS}
\setcounter{equation}{0}
Before going on, in the next section of the paper, to analyze various
possibilities for the nature of the metastate in the EA model, we
digress in this section to discuss the nature of metastates in some
much simpler models, where we do have some knowledge about the structure
of the pure phases. The purpose of this digression is not to increase
our understanding of these models, but rather to shed some light on
the general metastate approach. Consequently, in this section we
will primarily give heuristic arguments and not focus on results that
have been or can currently be proved rigorously. Indeed,
even though the models we will discuss are much simpler than
the EA model, in some of the cases, it appears that
obtaining a rigorous proof of
the conjectured nature of the metastate would be highly nontrivial.
All the models we will consider in this section
are nearest neighbor ones on the
usual lattice $\Z^d$. These are: a) homogeneous Ising ferromagnet,
b) random field Ising model (RFIM), c) highly disordered model of spin
glass ground states (see \cite{NS1994}, \cite{NS1996a}) and
d) homogeneous XY (i.e., two-component rotator) model. In cases a)
and d) where the Hamiltonians are not themselves disordered, we
will introduce disorder by means of random b.c.'s. It should be noted
that there already exist
some results, which implicitly or explicitly concern metastates,
in certain mean-field models, including in particular the
random field Curie-Weiss model \cite{AmPZ1992}, \cite{Ku1996}. Since
the mechanisms which control the nature of the metastate for
random field models seem to be essentially the same for nearest-neighbor
as for mean-field models, we will return to these mean-field results
below.
{\bf Homogeneous Ising Ferromagnet.} This is the standard Ising model
in which the all the nearest neighbor couplings have a constant positive
value, $\check{J}$.
At high temperature (at all nonzero temperatures, if $d=1$), there is
a unique infinite volume Gibbs state $P$ and so any sequence
of finite volume states (with $\Lambda_n$ tending to all of $\Z^d$)
converges (in the sense of finite dimensional distributions) to $P$.
In particular, this is so for the periodic b.c. states $P_L^{per}$
on cubes. Thus there is no CSD and the metastate is simply the
point measure supported on the single (necessarily pure) state $P$.
Let us now consider the $d=2$ model at low temperature (i.e., below
the critical temperature). Here, it has been proved \cite{A1980},
\cite{H1981} that there are exactly two pure infinite volume Gibbs
states, $P^+$ and $P^-$; these are the ones obtained with plus
and minus b.c.'s and thus are related by a global spin flip. It
follows that, since periodic b.c's do not break spin flip symmetry,
$P_L^{per}$ must converge to $P = (P^+ + P^-)/2$. Thus, with periodic
b.c.'s, there is again no CSD and the metastate is the point measure
supported on the single state $P$. The difference with
the high temperature
metastate is that $P$ is not pure but
rather a mixture of the two pure
states, $P^+$ and $P^-$. Free b.c.'s would behave the same as
periodic.
The situation becomes a bit more interesting
if we introduce disorder by taking random b.c.'s. I.e., we consider
the finite volume Gibbs states on cubes, $P_L^{\bar{s}(\omega)}$,
where for each L the b.c. spins at different sites on the
boundary are i.i.d. symmetric random variables and where we also
assume independence among the b.c.'s for different $L$'s. Heuristically,
we expect that (with probability close to $1$) $P_L^{\bar{s}(\omega)}$
is approximated by $W_{\omega ,L}^+ P^+ + W_{\omega ,L}^- P^-$ with
\begin{equation}\label{five1}
W_{\omega ,L}^{\pm} = e^{-\beta G_L^{\pm}(\omega)}/[
e^{-\beta G_L^+ (\omega)} + e^{-\beta G_L^- (\omega)}],
\end{equation}
where
\begin{equation}\label{five2}
G_L^{\pm}(\omega) = \pm \sum_{x \in \partial \Lambda_L}
(-\check{J}) \bar{s}_x
(\omega) .
\end{equation}
$G_L^{\pm}$ represents the interaction energy between the b.c.
spins and either the constant (identically $+1$ or identically
$-1$) spin configuration on $\Lambda _L$ or equivalently the
constant configuration on a single interior boundary layer
of spins at the sites $y \in \Lambda_L$ which are nearest
neighbors of $\partial \Lambda_L$. The asymptotic behavior of
$(W_{\omega ,L}^+ ,W_{\omega ,L}^- )$ is easy to see. By
the central limit theorem, $G_L^{\pm}/\sqrt{L}$ converges in
distribution to a normal random variable, and furthermore the
$G_L$'s are independent random variables for different $L$'s.
For almost every fixed $\omega$, $(W_{\omega, L}^+ ,
W_{\omega, L}^- )$ depends (extremely) chaotically on
$L$, taking values approaching $(1,0)$ for about half
the $L$'s and approaching $(0,1)$ for the rest; thus the
empirical distribution of $(W_{\omega,L}^+ ,W_{\omega,L}^-)$
for $L = 1,2, \dots , K$ converges to
$[\delta _{(1,0)} + \delta _{(0,1)} ]/2$. Hence we expect
that the metastate for random b.c.'s is the corresponding
$[\delta _{P^+} + \delta _{P^-} ]/2$
(and that there is an affirmative answer to the analogue
of Open Problem \ref{research49}).
Should there be any differences between $d \ge 3$ and
$d = 2$? The short answer is: {\it yes and no\/}.
A slightly longer answer is: {\it yes}, there are
of course differences, since for $d \ge 3$ and
sufficiently low temperature, the number of
pure states is infinite. But nevertheless, we expect
{\it no} difference between $d = 2$ and $d \ge 3$
in the nature of the metastates for periodic (or free)
and random b.c.'s. This is due to the nature of the
additional pure states, which are
non-translation-invariant
interface states obtained by taking
an infinite volume limit with Dobrushin b.c.'s (e.g.,
plus above or at the equator and minus below)
chosen so as to force an interface (between
configurations looking like the plus and minus states
far from the interface) at some plane perpendicular
to some coordinate direction (e.g., the equator).
For periodic or free b.c.'s, the finite volume states
converge to a translation invariant (and spin
flip symmetric) Gibbs state which (at least for
very low temperature) must be
$P = (P^+ + P^-)/2$ and so, as for $d = 2$,
the metastate is just $\delta _P$. Physically, one would
say that the weights of the pure states, other than
$P^+$ and $P^-$, in $P_L^{per}$ are suppressed because
these other states are much less consistent with
periodic b.c.'s. For random b.c.'s, the ratios
of the weight of $P^+$ to that of
$P^-$ should be, as for $d = 2$, of order
$e^{-G_L(\omega)}$ with $G_L / L^{(d-1)/2}$ random
and of order $1$, but the weights
of the other pure states should be suppressed
by extra factors of order $\exp (-L^{d-1})$ due
to the presence of the interface. Thus we still
expect the random b.c. metastate to be
$(\delta_{P^+} + \delta_{P^-})/2$.
(We note that essentially the same reasoning was used by
van Enter \cite{E1990} to argue the absence for $d=2$ of interface
ground states at zero temperature with random b.c.'s.)
{\bf Random Field Ising Model.} This model has
homogeneous ferromagnetic nearest neighbor couplings along
with independent random magnetic fields at each site.
The (formal) Hamiltonian is
\begin{equation}\label{five3}
-\check{J} \sum_{e=\{x,y\} \in \E^d} s_x s_y - \sum_{x \in \Z^d}
h_x(\omega) s_x ,
\end{equation}
where the $h_x$'s are i.i.d. symmetric random
variables on some probability space. For
$d=2$ and any $\beta$, there is a.s. a
unique infinite volume Gibbs state \cite{AW1990},
while for $d=3$ and sufficiently large $\check{J}$
and $\beta$ (with some assumptions on the common
distribution of the $h_x$'s), the plus and minus b.c.
infinite volume Gibbs states
$P_{\omega}^+ $ and $P_{\omega}^-$ are
a.s. distinct \cite{BriK1988} (see also \cite{I1985}).
Note that because of the
random field, these two states do {\it not}
map into each other under a global spin flip
and thus, for fixed $\omega$, $P_{\omega,L}^{per}$
should {\it not} converge to
$(P_{\omega}^+ + P_{\omega}^-)/2$. Here
we expect $P_{\omega,L}^{per}$ to be
approximated by
$W_{\omega,L}^+ P_{\omega}^+ + W_{\omega,L}^-
P_{\omega}^-$ with $W_{\omega,L}^{\pm}$ given as
in (\ref{five1}), but with (\ref{five2})
replaced by
\begin{equation}\label{five4}
G_L^{\pm}(\omega) = \pm \sum_{x \in \Lambda_L}-h_x(\omega) .
\end{equation}
For the RFIM, $G_L^{\pm}/L^{d/2}$ converges in distribution
to a normal random variable, but, unlike for the
homogeneous ferromagnet with random b.c.'s, the
$G_L$'s are not independent for different $L$'s.
As in the homogeneous case, but without the
need for random b.c.'s, we expect CSD to occur
and the metastate to be
$(\delta_{P_{\omega}^+} + \delta_{P_{\omega}^-})/2$.
Indeed, precisely this behavior was proved for the
Curie-Weiss model in a random field,
where the necessary approximations can be
controlled rigorously \cite{AmPZ1992},
\cite{Ku1996}. But it was also proved in
\cite{Ku1996} for the Curie-Weiss
random field model, that if a (sparse) subsequence of
volumes is not chosen, then a.s. the empirical
distribution does {\it not} converge to the
metastate. Heuristically, one expects essentially
the same phenomenon to occur for the short
range RFIM. The basic mechanism is that the
fraction of $L$'s between $1$ and $K$ for which
$G_L^+ (\omega) > 0$ does not converge
a.s. to $1/2$ as $K \to \infty$, but rather
converges in distribution to a random
variable with a continuous distribution
between $0$ and $1$ (related to the
arcsine law of Brownian motion). The reader is referred
to \cite{Ku1996} for more precise statements
and for proofs, in the Curie-Weiss context.
Nevertheless, the empirical distribution of
$(W_{\omega,L}^+ ,W_{\omega,L}^-)$, along a
(deterministic) subsequence
$L=L_1,\dots , L_K$ with $L_{j+1} -L_j$
growing rapidly enough, does converge
a.s. to $[\delta_{(1,0)} + \delta_{(0,1)}]/2$.
The subsequence should be chosen so that
for $j,j' \to \infty$ with
$j'/j \le \eta < 1$, $(L_{j'})^d/(L_j)^d \to 0$
in order that $G_{L_j}^+ /(L_j)^{d/2}$ be
asymptotically independent of
$G_{L_{j'}}^+/(L_{j'})^{d/2}$. This will guarantee
that the fraction of $j$'s between $1$ and $K$ for which
$G_{L_j}^+(\omega) > 0$ {\it does \/} converge a.s.
to $1/2$ as $K \to \infty$. For example,
constructing the subsequence by taking
$L_{j+1} = cL_j$ with a fixed $c > 1$
is much more than sufficient for this purpose.
{\bf Highly Disordered Spin Glass.} This is a
nearest neighbor Ising spin glass model (in
which the couplings depend [nonlinearly] on
the volume) that was introduced and analyzed in
\cite{NS1994}, \cite{NS1996a} (see also
\cite{CiMB1994}). At zero temperature,
the model has a quite interesting dimension
dependent ground state structure, while at strictly
positive temperature, the model makes little
or no thermodynamic sense. Before reviewing
the model and its ground state structure, we briefly
discuss the construction of metastates at
zero temperature, for models where the couplings
do not depend on the volume.
The ground states in a finite volume $\Lambda$
for b.c. * are the spin configurations $s$ in
${\cal S}_\Lambda$
that minimize the finite volume
Hamiltonian $H_{\Lambda}^*$. The natural
replacement, at zero temperature, for the
Gibbs distribution $P_{\Lambda, \beta}^*$ is
$P_{\Lambda, \infty}^* = \lim_{\beta \to \infty}
P_{\Lambda, \beta}^*$, the probability
measure which assigns equal probability to all the
ground states and zero probability to all
other spin configurations. Using $P_{\Lambda,
\infty}^*$, one can construct metastates
$\kappa _{\cal J}$ by the zero temperature
analogues of the lemmas and propositions of the
Appendix. The DLR equations for the $\Gamma$'s
in the support of $\kappa _{\cal J}$ are replaced
by the property that $\Gamma$ is supported
on infinite volume ground state configurations,
i.e., on spin configurations on $\Z ^d$ such
that the flip of any finite set of spins
yields a non-negative change in the energy.
It is worth noting that in the EA model, where
spin flip symmetry is broken only by (certain)
b.c.'s, if the common distribution $\mu$ of the
individual couplings is continuous, then
$H_{\Lambda}^*$ with a b.c. such as periodic
or free (resp., such as plus or random) a.s. has
exactly two ground states, related by a spin
flip (resp., one ground state). Thus the
$\Gamma$'s in the support of the metastate would all
be of the form $(\delta_s + \delta_{-s})/2$ (resp.,
$\delta_s$) for an infinite volume ground
state configuration $s$.
The highly disordered spin glass model of
\cite{NS1994} is one defined on the sequence of cubes
$\Lambda_L$, where the nearest neighbor Hamiltonians,
$H_{\Lambda_L}^*$ have couplings
$J_e^{(L)}$ depending nonlinearly on $L$
as follows.
\begin{equation}\label{five5}
J_e^{(L)}=\alpha_e \exp (-\lambda^{(L)}K_e ),
\end{equation}
where the $K_e$'s are i.i.d. continuous random variables, the
$\alpha_e$'s are i.i.d., independent of the $K_e$'s, with
$\nu(\alpha_e=+1)=\nu(\alpha_e=-1)=1/2$, and the $\lambda^{(L)}$'s are positive
constants. The $K_e$'s are not restricted as to sign and in fact can
have any continuous common distribution $\tilde{\mu}$, but
it is convenient to take $\tilde{\mu}$ to be the uniform
distribution on $(0,1)$. The
$\lambda^{(L)}$'s are
chosen so that each of the
$J_e^{(L)}$'s appearing in $H_{\Lambda_L}^{\bar{s}^{(L)}}$ has a
magnitude ``on its own scale''. More precisely, let $m_L$ denote the
number of edges $e=\{x,y\}\in \E^d$ with $x$ or $y$ (or both) in
$\Lambda_L$ and arrange the $m_L$ random variables $|J_e^{(L)}|$, for
these $e$'s, in rank order,
\begin{equation} \label{five6}
|J_{(1)}^{(L)}| \ge |J_{(2)}^{(L)}| \ge \dots \ge |J_{(m_L)}^{(L)}| ;
\end{equation}
$\lambda^{(L)}$ is chosen so that a.s., for all
large $L$,
\begin{equation}\label{five7}
|J_{(j)}^{(L)}| \ge 2 |J_{(j+1)}^{(L)}| \quad \mbox{ for } 1 \le j < m_L.
\end{equation}
This immediately implies that a.s., for all large $L$,
\begin{equation}\label{five8}
|J_{(j)}^{(L)}| > \sum_{k=j+1}^{m_L} |J_{(k)}^{(L)}| \quad \mbox{ for } 1 \le
j < m_L.
\end{equation}
For $L$ large enough so that (\ref{five8})
is valid, the finite volume ground states can be determined
in terms of certain finite volume tree
graphs depending only on the $K_e$'s and on
the type of b.c. We will restrict
attention to fixed and free b.c.'s. For
fixed b.c.'s, we construct a random graph
$F^{(L)}$ which is a subset of the nearest
neighbor graph on the sites of
$\Lambda_L \cup \partial \Lambda_L$.
The set of edges in $F^{(L)}$ is
defined inductively as follows. (The vertices
in $F^{(L)}$ are all sites in
$\Lambda_L \cup \partial \Lambda_L$ touching
these edges). Begin with the edge (among
those with at least one endpoint in
$\Lambda_L$) with the smallest value
of $K_e$ (i.e., the one corresponding to
$J_{(1)}^{(L)}$), then sequentially add
edges with the smallest values of $K_e$
among the remaining edges, except not any
edge that would either create a closed loop
or would cause some site in $\Lambda_L$ to be
connected to more than one site in
$\partial \Lambda_L$. Stop when every site
in $\Lambda_L$ is connected to exactly one
site in $\partial \Lambda_L$. $F^{(L)}$ is a
forest (i.e., a union of site-disjoint trees) such that
each site $x \in \Lambda_L$ is connected to
a unique site $w_L(x)$ in $\partial \Lambda_L$
by a unique path $r_L(x)$ in $F^{(L)}$.
The fixed b.c. ground states are given as
follows. Define
\begin{equation}\label{five9}
\eta_L(x) = \prod_{e \in r_L(x)} \alpha_e.
\end{equation}
Then the ground state $s^{(L)}$ for $H_{\Lambda_L}^{\bar{s}^{(L)}}$ is
given by
\begin{equation}\label{five10}
s_x^{(L)}=\eta_L(x) \bar{s}_{w_L(x)}^{(L)}, \mbox{ for } x \in
\Lambda_L.
\end{equation}
For free b.c.'s, $F^{(L)}$ is replaced by
$F^{(L)f}$, a subgraph of the nearest
neighbor graph on $\Lambda_L$ in which the edges
with smallest $K_e$ are sequentially added,
with no loop creation allowed, until the
graph becomes a single tree touching every
site in $\Lambda_L$. The pair of free b.c.
ground states, $\pm s^{(L)f}$, is
determined by the requirement that
\begin{equation}\label{five11}
s_x^{(L)f}s_{x'}^{(L)f} =
\eta_L(x,x')=\prod_{e\in r_L(x,x')} \alpha_e ,
\end{equation}
where, for $x,x' \in \Lambda_L$,
$r_L(x,x')$ is the unique path in
$F^{(L)}$ connecting $x$ and $x'$.
The set of all (subsequence) limits $s$ of fixed
b.c. ground states $s^{(L)}$, for all possible
choices of b.c. $\bar{s}^{(L)}$, is completely
characterized in terms of $F_\infty$ (the limit
of $F^{(L)}$ or of $F^{(L)f}$ as
$L \to \infty$), which is a forest,
spanning all of
$\Z^d$, each of whose trees is infinite.
Namely, for each tree in
$F_{\infty}$, the relative signs of the spins
in $s$ on that tree are determined by the analogue
of (\ref{five11}), with the absolute sign
of a single spin in each tree undetermined. The
number of such infinite volume configurations is
thus $2^{\cal M}$, where ${\cal M}$ is the (a.s.
constant) number of trees in the forest
$F_{\infty}$. This forest is a natural object
in the context of invasion percolation
(see \cite{NS1994}, \cite{NS1996a} for further
discussion and for historical references)
and in that context, it was proved rigorously
that ${\cal M} = 1$ for $d = 2$ \cite{CCN1985}.
It has been conjectured \cite{NS1994}, \cite
{NS1996a} but not proved that ${\cal M} =1$ for
$d < 8$ and that ${\cal M} = \infty$ for
$d > 8$.
When ${\cal M}=1$, the metastate structure is like
that for the homogeneous Ising ferromagnet. For
free b.c.'s, there will be no CSD and the
metastate will be the point measure supported on
$\Gamma = (\delta_s + \delta_{-s})/2$ where
$\pm s$ is the pair of infinite volume ground states.
For plus b.c.'s, the behavior will be like that for the
random b.c. homogeneous ferromagnet, i.e.,
the metastate will be $(\delta_{\delta_s} +
\delta_{\delta_{-s}})/2$. Note in this regard
that for plus b.c.'s, the ground state spins,
$s_x^{(L)}$ at a fixed $x$ as $L$ varies are i.i.d.
symmetric random variables (in their dependence
on the $\alpha_e$'s for fixed $K_e$'s), because
each time L is increased, at least one new $\alpha_e$
appears in (\ref{five9}).
When ${\cal M} = \infty$, similar reasoning
shows that with plus b.c.'s, the signs of
$s_{x_1}^{(L)},\dots ,s_{x_j}^{(L)}$ for
fixed $x_1, \dots ,x_j$ in distinct trees
are i.i.d. symmetric random variables for
fixed $L$ and independent as $L$ varies.
Here, the metastate corresponds to the measure
supported on the infinite volume configurations
$s$ corresponding to independent (fair) coin
tosses that determine the overall sign of each
of the trees in $F_\infty$.
When ${\cal M} = \infty$, but free b.c.'s
are used, the situation is trickier. For
each $L$, the edges of $F^{(L)f}$ consist
of those edges of $F_\infty$ within
$\Lambda_L$ plus enough extra edges (with the
smallest posible $K_e$ values) to form a
{\it single} tree. These edges are known
to run off to infinity as $L \to \infty$; this
gives asymptotic independence (between
$L$ and $L'$) of the relative
signs of the trees touching fixed locations,
as determined by the signs of these extra
edges, providing $L'$ is sufficiently larger
than $L$. Thus the metastate will be supported on
$\Gamma$'s of the form
$\Gamma = (\delta_s + \delta_{-s})/2$,
with
the metastate measure on the pairs
$\{s,-s\}$ corresponding to independent coin
tosses to determine the relative signs between
different trees. However, since the choices for
$L$ and for $L+1$ are (unlike in the plus b.c.
case) not independent, it is a priori possible that
a phenomenon like that found in \cite{Ku1996}
for the random field Curie-Weiss model could occur;
i.e., it could be that the empirical distribution
over $L=1,2,\dots ,K$ would not converge a.s. to
the metastate. We believe however, based on heuristic
arguments, that this is not so, although we have
not worked out the details of a complete proof.
We conclude our discussion of this model by noting
that the situation for periodic b.c.'s
(when ${\cal M} = \infty$) seems
more difficult to analyze than for
free b.c.'s.
{\bf Homogeneous XY Model.} In this model, the spin
variables $s_x = (s_x^1,s_x^2)$ take values in the
unit circle and the (formal) Hamiltonian is
\begin{equation}\label{five12}
-\check{J} \sum _{\{x,y\} \in \Z^d}
(s_x^1 s_y^1 + s_x^2 s_y^2) .
\end{equation}
We take $d=3$, since in that dimension, at low
temperature, there exist distinct infinite volume
Gibbs states $P^\theta$ obtained by taking fixed
constant b.c.'s, in which $\bar{s}$ is chosen
by setting $\bar{s}_x$ to be
$(\cos \theta, \sin \theta)$ for every $x$ in
$\partial \Lambda_L$ \cite{FSS1976}.
We will assume, for the purposes
of this discussion, that the set of all
pure infinite volume Gibbs states is exactly
$\{P^\theta :0 \le \theta < 2\pi \}$.
If one takes free or periodic b.c.'s, which
do not break the spin rotation symmetry, then
there would be no CSD and the finite volume Gibbs
states would converge to $P = \int_0^{2 \pi}
P^\theta d \theta$. The metastate is then
$\delta_P$,
analogously to the
homogeneous Ising model (where
$P = (P^+ + P^-)/2$).
On the other hand, the situation is less clear if we take
random b.c.'s. I.e., suppose for each $L$,
the b.c. $\bar{s}(\omega)$ has i.i.d.
$\bar{s}_x$'s for $x \in \partial \Lambda_L$
(each with a uniform distribution on the unit
circle) and we also take the b.c.'s for different
$L$'s to be independent. Now, for each fixed
$\omega$, the rotation symmetry is broken, although
the ensemble of b.c.'s is of course (statistically)
rotation invariant. There are (at least) two
natural a priori guesses for the metastate in
this case: 1. The ``competition'' among pure states
for the random b.c.'s is so balanced that we end up with
the same metastate as for free or periodic b.c.'s,
i.e., $\delta_P$ with $P$ the uniform mixture
of the pure states $P^\theta$ over $\theta$, as
above. 2. The competition is so unbalanced that for
large $L$, essentially one $\theta = \theta_L (\omega)$
wins out (decisively) with that $\theta_L$ distributed
uniformly (as a function of $\omega$ for fixed $L$)
and depending chaotically on $L$ (for fixed $\omega$);
the metastate will then be supported on $\Gamma$'s
each of which is one of the pure states
$P^{\theta}$, with a uniform distribution of the
$\theta$'s.
In fact, we conjecture that what actually
happens is case 2, based on the following heuristic
arguments modelled after the ones we used above for
the homogeneous Ising ferromagnet.
We expect that $P_L^{\bar{s}(\omega)}$ is approximated
by $\int_0^{2\pi} W_{\omega,L}^{\theta} P^{\theta} d\theta$
with
\begin{equation}\label{five13}
W_{\omega,L}^{\theta} = e^{-\beta G_L^\theta (\omega)}/
\int_0^{2\pi} e^{-\beta G_L^\theta (\omega)} d \theta ,
\end{equation}
where
\begin{equation}\label{five14}
G_L^{\theta}(\omega) = \sum_{x \in \partial \Lambda_L}
(-\check{J})(\bar{s}_x^1 (\omega) \cos \theta +
\bar{s}_x^2 (\omega) \sin \theta) .
\end{equation}
The metastate is supported on states
$\Gamma$ of the form
$\int_0^{2\pi}P^{\theta} \rho(d \theta)$,
where $\rho$ is a probability measure on
$[0,2\pi)$, with some distribution of
$\rho$'s. With our ansatz for approximating
$P_L^{\bar{s}}(\omega)$, this distribution
would be the limiting distribution of
the sequence of random measures
$\rho_L (d\theta) = W_{\omega,L}^{\theta} d \theta$.
What is this limit in distribution?
By using the central
limit theorem for the random vectors,
$\sum_{x \in \partial \Lambda_L}({\bar{s}}_x^1,{\bar{s}}_x^2)$,
one can prove that $W_{\omega,L}^{\theta} d \theta$
does converge in distribution to the random point
measure $\delta_{\theta'}$ with $\theta'$
uniformly distributed on $[0,2\pi)$. Although
this supports our conjecture that case 2 is the
correct guess, it does not rigorously prove the
conjecture, since the connection between
$P_L^{\bar{s}}$ and $W_{\omega,L}^{\theta}$ is
an ansatz. Obtaining a complete proof
is an interesting open problem.
\section{Metastates in the Edwards-Anderson Model}
%\markboth{CH. 6. METASTATES IN THE EA MODEL}{CH. 6. METASTATES IN THE EA MODEL}
\setcounter{equation}{0}
We proceed, as in \cite{NS1995a}, by giving a partial classification of
the possible types of metastates $\kappa_{\cal J}$ which could
occur in the EA model. The simplest of these, and one
which of course does occur, at least for
small $\beta$ or for $d=1$ (as in the homogeneous ferromagnet),
is possibility 1) that $\kappa_{\cal J}$ is
(for $\bar{\nu}$-a.e. ${\cal J}$) supported on a single pure Gibbs state
$P=P_{\cal J}$. This is the case, for example, if ${\cal N}={\cal
N}(d,\mu,\beta)=1$
(e.g., for $\beta<\beta_c^*(d,\mu)$),
since then $P_{{\cal J},L}\to P_{\cal J}$ as $L\to \infty$.
It is important to note however, that this possibility could occur (at
least, in principle) even if ${\cal N} \ne 1$; indeed, just
such a situation of ``weak uniqueness'' (see
\cite{COE1987} and \cite{EnF1985})
happens in very long range spin glasses at
high temperature
\cite{FZ1987}, \cite{GNS1993}.
Another
simple possibility 2) is that $\kappa=\kappa_{\cal J}$ is supported on a
single Gibbs state, which is a mixture of two distinct pure states:
\begin{equation}\label{four33}
\kappa=\delta_P,\quad P=\frac{1}{2}P'+\frac{1}{2}P'',
\end{equation}
where $P'=P_{\cal J}'$ and $P''=P_{\cal J}''$ are pure states that are
global flips of each other. As in the $d=2$ homogeneous Ising
ferromagnet, this would be the case
according to the Fisher-Huse (FH) scaling picture discussed earlier,
which predicts that (when ${\cal N}(d,\mu,\beta)\neq 1$) such a $P_{\cal
J}'$ and $P_{\cal J}''$ are the only pure states so that (by obvious
spin flip symmetry arguments) $P_{{\cal J},L}\to \frac{1}{2}P_{\cal
J}'+\frac{1}{2}P_{\cal J}''$ as $L\to \infty$.
(We deal here [and in the rest of the paper, unless
otherwise noted] with $\kappa_{\cal J}$ constructed,
as in Sect. 4, with periodic b.c.'s. We note that the
same analysis should apply to other b.c.'s
that are spin-flip symmetric.) Of course, when $P_{{\cal
J},L}$ converges to some $P_{\cal J}$, there is no CSD and no need for a
metastate description. When the metastate is supported on a single
$P_{\cal J}$, we will say that it is not dispersed. Again, it is
important to note that possibility 2 can occur even if
${\cal N} > 2$, like in the $d=3$ homogeneous Ising ferromagnet
at low temperature (below the roughening temperature). That would
be an example of a ``weak scaling'' picture.
A trivial (i.e., over only two states) sort of dispersal,
analogous to what happens in the homogeneous ferromagnet
with random b.c.'s,
should occur in the scaling picture if one
replaces periodic boundary conditions by b.c.'s which break the spin-flip
symmetry (but still do not depend on $J$) such as random b.c.'s.
or more simply plus b.c's. Here, one expects the
finite volume state to exhibit CSD, approximating $P'$ for (roughly)
half of the $L$'s and $P''$ for the other half so that the metastate
(for such a b.c.) should be
$\frac{1}{2}\delta_{P'}+\frac{1}{2}\delta_{P''}$, which is quite
different from (\ref{four33}).
The phenomenon observed by K\"ulske
(in the random field Curie-Weiss model) \cite{Ku1996}, that
subsequences of volumes are essential in order to have a.s.
convergence of the empirical distribution of the metastate,
should not occur in the context of this possibility, but
conceivably could in some other possibilities we discuss below.
Other sorts of dispersal could happen if there are more then two
pure states, as in an SK picture. Roughly speaking, according to
Theorem \ref{theo410}, a dispersed metastate describes the way in which,
for large $L$,
\begin{equation}\label{four34}
P_{{\cal J},L}\approx \sum_\alpha W_{{\cal J},L}^\alpha P_{\cal
J}^\alpha,
\end{equation}
when there is CSD and $P_{{\cal J},L}$ does {\it not} converge to a
single $P_{\cal J}$.
In possibilities 1 and 2 that we discussed above, most of the weight
(represented by the $W_{{\cal J},L}^\alpha$'s) is concentrated on one or
two pure states as $L\to \infty$. Two other possibilities (numbers 5 and
6, the
latter of which includes the nonstandard SK picture) that we will soon discuss
also have the property that most weight
in a given volume is concentrated on a few states,
even though there are (uncountably) many pure states. But first, we
briefly mention two possibilities (very different from either FH or SK
pictures) where the weights are shared more equitably. In both of these
possibilities $\kappa_{\cal J}$ is supported on infinite volume Gibbs
states $\Gamma$ whose decomposition into pure states is
continuous. Possibility 3) is that there is no dispersal and
$\kappa_{\cal J}$ is supported on a single such $\Gamma$, while
possibility 4) is that there is dispersal and $\kappa_{\cal J}$ is
supported on multiple such $\Gamma$'s. As there seems to be no
particular reason to suppose that either of these possibilities occurs
in the EA model, we proceed to the next possibility, which we find
rather intriguing.
Possibility 5) is one in which there is CSD and dispersal among (uncountably)
many $\Gamma$'s, but each of these $\Gamma$'s decomposes into just two pure
states related by a global flip, like in possibility 2. This possibility,
where ${\cal N}=\infty$ but for each large $L$ one ``only sees two
pure states'' (in the sense that $P_{{\cal J},L}$ is approximately a
mixture of just two
pure states) is intermediate between the scaling picture
where ${\cal N}=2$ (or more generally, where possibility 2 occurs
even if ${\cal N} > 2$) and the
next possibility (which leads to the nonstandard SK picture) where
${\cal N}=\infty$
and for large $L$ one ``sees many pure states''. In the context
of possibility 5, it would appear that there is a chance for
a phenomenon like the one seen in \cite{Ku1996} for the
random field Curie-Weiss model to occur. This would mean, roughly
speaking, that the pair of pure states seen for a given large
$L$ would tend to be the same pair of states seen for $L' > L$ until
$L'/L$ became large. Before leaving possibility 5, we note that
its analogue for the situation where periodic b.c.'s are replaced
by plus or by random b.c.'s would be one with a metastate
supported on $\Gamma$'s that are single pure states chosen from the
continuum of all pure states. This would be analogous to the behavior
conjectured in the last section for the homogeneous
$XY$ model with random b.c.'s.
Possibility 6) has, like possibility 5, nontrivial dispersal over many
$\Gamma$'s, but unlike possibility 5, its $\Gamma$'s have a nontrivial
decomposition into pure states $-$ namely
\begin{equation}\label{four35}
\Gamma =\sum_\alpha W_\Gamma^\alpha P_{\cal J}^\alpha~;
\end{equation}
this decomposition should be discrete (i.e., a countable sum, as indicated)
but with many (in particular, more than two) nonzero weights $W_\Gamma^\alpha$.
As we shall see, the nonstandard SK picture will require that the dispersal
of $\kappa_{\cal J}$ over $\Gamma$'s of the form (\ref{four35}) is a dispersal
not only over the weights $W_\Gamma^\alpha$ but also over the selection of
the countably many pure states appearing in the discrete sum (from an
uncountable family of all pure states for the given ${\cal J}$). To explain
why that is so, we first need to discuss replicas and their overlaps within
the metastate framework. (We remark, as previously discussed
in the context of the standard SK picture, that, strictly speaking, it
is the overlaps which must be discrete rather than the pure state
decomposition.)
\section{Replicas and Overlaps}
%\markboth{CH. 7. REPLICAS AND OVERLAPS}{CH. 7. REPICAS AND OVERLAPS}
\setcounter{equation}{0}
Replicas and overlaps are a way of probing the meaning of the approximate
equation (\ref{four34}), our EA model replacement for the SK approximate
equation (\ref{four16}). It is therefore natural to take replicas for fixed
$L$ and only afterward let $L\to\infty$. As emphasized by Guerra \cite{G1995},
this order of operations could yield a different result than that
obtained by first letting $L\to
\infty$ and then taking replicas, as was done in the standard SK picture.
In other words, rather than letting $L\to\infty$ first to obtain a single
infinite volume state $P_{\cal J}$ and then defining replicas $s^1,s^2,s^3,
\dots$ by using the product measure $P_{\cal J}(s^1)\times P_{\cal
J}(s^2)\times \dots$,
we will first take the product measure for finite volume, $P_{{\cal J},L}(s^1)
\times P_{{\cal J},L}(s^2)\times\dots$, and then let $L\to\infty$.
We have already encountered three (related) ways to consider the limit
as $L\to\infty$ of $P_{{\cal J},L}$ (without replicas). These are: (a) to
obtain $P_{\cal J}$ via the limit of the joint distribution ($\bar{\nu}
({\cal J})\times P_{{\cal J},L}(s)$) of $J$ and $S$ (see the
remark following Theorem \ref{prop47}), (b) to
obtain $\kappa_{\cal J}$ via the limit of the joint distribution of $J$
and $P_{J,L}$ (see Theorem \ref{prop47}) and (c) to obtain $\kappa_{\cal J}$
via the limit for fixed ${\cal J}$ of the empirical distribution of
$P_{{\cal J},L}$
(see Theorem \ref{theo410}). We want to see what happens to these three
types of limits when replicas are taken first, i.e., when $P_{{\cal J},L}$
is replaced by the infinite product measure (on ${\cal S}_{\Lambda_L}
\times {\cal S}_{\Lambda_L}\times \dots$),
\begin{equation}\label{four36}
P_{{\cal J},L}^\infty (s^1,s^2,\dots)=P_{{\cal J},L}(s^1)\times
P_{{\cal J},L}(s^2)\times \dots,
\end{equation}
which is the finite volume Gibbs state for (arbitrarily many) replicas. The
next proposition (the first part of which is explicit in \cite{NS1995a}
with the rest implicit) gives the answer, which is that $\kappa_{\cal J}
(\Gamma)$ is replaced by $\kappa_{\cal J}^\infty$, the probability measure
on $\Omega^1\times \Omega^1 \times \dots$, supported on elements of the
form $\Gamma \times \Gamma \times \dots$, with $\Gamma$ distributed by
$\kappa_{\cal J}$. Convergence of measures or random variables in the
proposition denotes, as usual, (weak) convergence of finite
dimensional distributions.
\begin{prop}[\cite{NS1995a}]\label{prop411} Let $L_n$, $n_k$ and $K_m$
be as in Theorems \ref{prop47} and \ref{theo410}. Then
\begin{equation}\label{four37}
\bar{\nu}({\cal J})\times P_{{\cal J},L_n}^\infty \to
\bar{\nu}({\cal J})\times P_{{\cal J}}^\infty \quad \mbox{ as } n \to
\infty,
\end{equation}
where $P_{{\cal J}}^\infty$ is the measure on ${\cal S}^\infty$,
\begin{equation}\label{four38}
P_{{\cal J}}^\infty(s^1,s^2,\dots)= \int \left[ \Gamma(s^1) \times
\Gamma (s^2)\times \dots \right] d\kappa_{\cal J}(\Gamma);
\end{equation}
\begin{equation}\label{four39}
(J,P_{J,L_n}^\infty) \to \bar{\nu}({\cal J})\times \kappa_{\cal J}^\infty \quad
\mbox{ as } n\to \infty;
\end{equation}
and finally, defining the empirical distribution for replicas to be
the measure on $\Omega^1\times\Omega^1\times\dots$,
\begin{equation} \label{four40}
\kappa_J^{K,\infty} \left( (L_n) \right) =
\frac{1}{K} \sum_{k=1}^K \delta_{(\vec{Q}^{(L_k)},\vec{Q}^{(L_k)},
\dots )},
\end{equation}
we have
\begin{equation}\label{four41}
\mbox{ for $\nu$-a.e. }\omega,\quad
\kappa_{J(\omega)}^{K_m,\infty} \left( (L_{n_k}) \right) \to
\kappa_{J(\omega)}^\infty \quad \mbox{ as } m\to \infty.
\end{equation}
\end{prop}
{\it Proof.} We sketch the proofs in the reverse order of the
proposition. The last claim (\ref{four41}) easily reduces to the following
fact, whose proof we leave to the reader: if $\vec{Q}_l$ is a sequence of
random vectors whose sequence of distributions (on
$\R^M$, for some $M$) converges as $l\to\infty$ to some $\rho$, then
the sequence
of distributions of $(\vec{Q}_l,\vec{Q}_l,\dots,\vec{Q}_l)$ (on $(\R^M)^k)$)
converges to $\rho\times\rho\times \dots \times \rho$. Similarily, the middle
claim (\ref{four39}) reduces to: if $(\vec{J}_l,\vec{Q}_l)$ converges in
distribution to some $\rho^\dagger$ whose marginal
distribution (of $\vec{q}$, given
$\vec{{\cal J}}$) is $\rho_{\vec{{\cal J}}}$, then $(\vec{J}_l,\vec{Q}_l,
\dots,\vec{Q}_l)$ converges to a measure whose marginal
distribution (of $(\vec{q},\dots,
\vec{q})$, given $\vec{{\cal J}}$) is $\rho_{\vec{{\cal J}}}\times \dots \times
\rho_{\vec{{\cal J}}}$. The first claim (\ref{four37}) follows from (\ref{four39})
by the replica version of the remark following Theorem
\ref{prop47}. This is essentially the same as
arguing directly from Theorem \ref{prop47} by using ``metacorrelations'',
as in \cite{NS1995a}. That argument makes clear that replicas are
already implicit in the very notion of metastates, since the natural
generalization of (\ref{four27}) beyond functions linear in $q$ replaces
$\langle S_A \rangle_{J,L_n}$ by the metacorrelation
\begin{equation}\label{four42}
\langle S_{A_1} \rangle_{J,L_n} \cdots
\langle S_{A_m} \rangle_{J,L_n} = \langle S_{A_1}^1 \cdots S_{A_m}^m
\rangle_{J,L_n}^\infty,
\end{equation}
where the latter bracket denotes the expectation with respect to
$P_{J,L_n}^\infty$.
\EndProof
\setcounter{theo}{2}
{\bf Remark 7.2} There are two conceptually important points hidden in
the last proposition. One, which is the essential content of (\ref{four41}),
is that replicas must be described in the infinite volume limit by
$\Gamma \times \Gamma \times \dots$ with $\Gamma$ distributed by
$\kappa_{\cal J}$ because replicas in finite volume are taken from
the {\it same} $L$ and $\kappa_{\cal J}$ (according to Theorem \ref{theo410})
describes the sampling of states as $L$ varies. The second, emphasized
by Guerra \cite{G1995}, is that in the infinite volume replica measure
$P_{\cal J}^\infty$, the replicas need not be independent, even though they
of course are independent (by construction) in the finite volume measure
$P_{{\cal J},L}^\infty$.
Indeed, from the form of (\ref{four38}), we see that they will only be
independent if $\kappa_{\cal J}$ is non-dispersed. In \cite{NS1995a},
this lack of independence when the metastate is dispersed is called
replica non-independence (RNI). It is equivalent to the non-interchangeability
of the operations of taking replicas and letting $L\to \infty$, since done
in the other order (as in the standard SK picture) the measure obtained is
\begin{equation}\label{four43}
P_{\cal J}(s^1) \times P_{\cal J}(s^2)\times \dots
= \int \Gamma_1(s^1)d\kappa_{\cal J}(\Gamma_1) \times
\int \Gamma_2(s^2)d\kappa_{\cal J}(\Gamma_2) \times \dots ,
\end{equation}
which does not agree with $P_{\cal J}^\infty$ unless $\kappa_{\cal J}$
is non-dispersed.
Now that we have seen that replicas are described by $\kappa_{\cal J}^\infty$,
it is fairly clear what the appropriate replacements are for the overlap
$R_{\cal J}$ and overlap distribution ${\cal P}_{\cal J}$ as defined
(in the standard SK picture) by using the two-replica product measure
$P_{\cal J}
\times P_{\cal J}$. Namely, on ${\cal S}\times{\cal S}$ with product measure
$\Gamma \times \Gamma$, we define the random variable
\begin{equation}\label{four44}
R_\Gamma = \lim_{L\to\infty}|\Lambda_L|^{-1} \sum_{x\in\Lambda_L} s_x^1
s_x^2,
\end{equation}
and define ${\cal P}_\Gamma$ to be (the formal density of) its distribution.
The nature of ${\cal P}_\Gamma$ depends of course on the nature of $\Gamma$,
and more particularly on its pure state decomposition. Thus if $\Gamma$ is
a pure state (and the limit in (\ref{four44}) exists), $R_\Gamma$ will be
the constant $r_\Gamma$ in which $s_x^1s_x^2$ in (\ref{four44}) is replaced
by the expectation $\langle s_x \rangle_\Gamma^{~2}$; if $\Gamma$ is pure and
flip invariant then $\langle s_x \rangle_\Gamma\equiv 0$ and ${\cal P}_\Gamma
=\delta(r)$. If $\Gamma$ has the decomposition $\Gamma=\frac{1}{2} P'
+\frac{1}{2} P''$ with $P'$ and $P''$ related by a global flip (as in
possibilities 2 and 5 discussed in the previous section),
then ${\cal P}_\Gamma =
\frac{1}{2} \delta(r-r_0) + \frac{1}{2} \delta(r+r_0)$, where $r_0= r_{P'} =
r_{P''}$. If $\Gamma$ is as in (\ref{four35}) of possibility 6, then
${\cal P}_\Gamma$ has the SK type form (see (\ref{four18})),
\begin{equation}\label{four45}
{\cal P}_\Gamma = \sum_\alpha \sum_\gamma W_\Gamma^\alpha W_\Gamma^\gamma
\delta (r-r_{\alpha\gamma}).
\end{equation}
It is important to note that our analysis of the relation between
the state $\Gamma$ and the overlap distribution
${\cal P}_\Gamma$ is based on the definition
(\ref{four44}) for the overlap. The limit in (\ref{four44})
is taken within an already infinite system, corresponding
physically to taking the overlap in a box which is large
but much smaller than the overall system size. Were one to
take the overlap box and system size the same, one might,
in principle, obtain a different limiting distribution.
For example, Huse and Fisher \cite{HuF1987} noted that in the
$d=2$ homogeneous Ising ferromagnet, such a limit (with
antiperiodic b.c.'s) yields a {\it continuous\/} overlap
distribution (rather than $\frac{1}{2} \delta(r-r_0) +
\frac{1}{2} \delta(r+r_0)$) despite the presence of only
two pure states.
In the metastate approach, even with ${\cal J}$ fixed, $\Gamma$ should itself
be regarded as random with the distribution $\kappa_{\cal J}(\Gamma)$.
Thus ${\cal P}_\Gamma$ should be regarded as random with the distribution
induced on it by $\kappa_{\cal J}$. In possibilities 1 and 2 (and 3),
$\kappa_{\cal J}$ is non-dispersed, so ${\cal P}_\Gamma$ is non-random.
In possibility 5, $\kappa_{\cal J}$ is dispersed, but if we postulate (as
in the SK pictures) that $r_{\alpha\alpha}$ (see (\ref{four17})) is the same
($=r_{EA}$) for all pure states $P_{\cal J}^\alpha$, then ${\cal P}_\Gamma$
is again non-random. In possibility 6, $\kappa_{\cal J}$ is dispersed and
${\cal P}_\Gamma$ should really be random, i.e., should depend on $\Gamma$
since the $W_\Gamma^\alpha$'s and the selection of the
$r_{\alpha\gamma}$'s that have nonzero weights should depend
on $\Gamma$.
\section{The Nonstandard SK Picture}
%\markboth{CH. 8. THE NONSTANDARD SK PICTURE}{CH. 8. THE NONSTANDARD SK PICTURE}
\setcounter{equation}{0}
The dependence of ${\cal P}_\Gamma$ on $\Gamma$ (for fixed ${\cal J}$),
just discussed in the previous section of the paper, is crucial for the
nonstandard SK picture because there can be no dependence on ${\cal J}$.
More precisely, if one considers the distribution of the random $P_\Gamma$
(that it inherits from $\kappa_{\cal J}$) as a function of ${\cal J}$,
there is a translation invariance/ergodicity argument, essentially the same
as in Proposition \ref{prop45} and the discussion preceding it, which
shows that that distribution is self-averaged. Such a self-averaging property
killed the standard SK picture. It does not kill the nonstandard picture
(at least, not on its own)
because there {\it dependence on ${\cal J}$ is replaced by dependence on
$\Gamma$ for fixed ${\cal J}$.} In particular $\bar{{\cal P}}=\int {\cal P}_{\cal J}
d\bar{\nu}({\cal J})$ is now replaced by
\begin{equation}\label{four46}
\tilde{{\cal P}} = \int {\cal P}_\Gamma d\kappa_{\cal J}(\Gamma).
\end{equation}
The nonstandard SK picture thus includes the following requirements
within the context of possibility 6: (1) ${\cal P}_\Gamma$ does depend
on $\Gamma$ as $\Gamma$ varies according to $\kappa_{\cal J}$; (2)
${\cal P}_{\Gamma}$ is (for
$\kappa_{\cal J}$-a.e. $\Gamma$) a sum of (countably) infinitely many
delta functions; (3) $\tilde{{\cal P}}$ is continuous (except for two
$\delta$-functions at $\pm r_{EA}$, whose weights add up to less than
one); and (4) ultrametricity of $\{r_{\alpha\gamma}: W_\Gamma^\alpha
W_\Gamma^\gamma \neq 0 \}$ (for
$\kappa_{\cal J}$-a.e. $\Gamma$).
Before going on to discuss the prospects for the validity of
the nonstandard SK picture, we point out that replica symmetry breaking (RSB),
which plays an important role in the usual analyses of the SK model, has
a very natural interpretation within the metastate framework. Replica
symmetry just refers to permutation invariance among the replica
labels. To be broken, there must be something which distinguishes
between the replicas, such as nontrivial $q^{\alpha\gamma}$'s with
$\alpha$ and $\gamma$ coming from different replicas, or more generally
different pure states coming from different replicas. Thus we consider
RSB to occur whenever the $\Gamma$'s in the support of the metastate are
not pure states so that when $\Gamma\times\Gamma\times\dots$ is decomposed
into pure states, different $\alpha$'s and $\gamma$'s will appear for
different replicas. In the physics literature, it is conventional to
regard situations such as possibility 2, where the replicas appear in
pure states which only differ by a global flip, as a trivial example of
RSB. Thus we would describe possibility 2 as having no dispersal of the
metastate and only trivial RSB while possibility 5 has dispersal of the
metastate (along with CSD and RNI) but still only trivial RSB. The
nonstandard SK picture has both dispersal of the metastate and
nontrivial RSB.
Should there be an affirmative answer to Open Problem
\ref{research45}, i.e., should it be shown that the number of pure
states (sometimes) exceeds 2, it would remain to determine whether the many
states are ``seen'' in $P_{{\cal J},L}$ for large $L$ and if so, how.
The nonstandard SK picture is only one of a number of ways in which the
many states might be seen, but it appears to be the only way in which
the essential features (suitably reinterpreted) of Parisi's mean-field
analysis would be present. It is an open problem to answer the
question: is
the nonstandard SK picture valid for any $d$, $\mu$ and
$\beta$?
\begin{research}\label{research413}
Determine whether for some $d$, $\mu$ and $\beta$, the metastate
$\kappa_{\cal J}$ is (for $\bar{\nu}$-a.e. ${\cal J}$) supported on (a)
many $\Gamma$'s, (b) $\Gamma$'s which decompose into more than two pure
states, (c) $\Gamma$'s which satisfy the other requirements of the
nonstandard SK picture.
\end{research}
What are the prospects for the nonstandard SK picture to be
valid? In discussing this question, we will focus on a
certain property of the metastate that already appears
in \cite{AW1990}; namely, covariance with respect to changes
of finitely many couplings. Let us assume, for simplicity,
that the common distribution $\mu$ of the individual
couplings is continuous with a strictly positive density
on all of $(-\infty,\infty)$, as in the usual Gaussian EA
model. Consider changing ${\cal J}$ to
${\cal J}+ \Delta {\cal J}$ where
$\Delta {\cal J}$ is nonzero on some finite
collection of edges of $\E^d$ and further consider the
corresponding map $\phi_{\Delta {\cal J}}$ from
infinite volume Gibbs states $\Gamma$ for ${\cal J}$
to those, $\Gamma' = \phi_{\Delta {\cal J}}(\Gamma)$,
for ${\cal J}+ \Delta {\cal J}$ (all at a given
$\beta$) defined by
\begin{equation}\label{eight2}
\langle s_A \rangle_{\Gamma'} =
\langle s_A e^{-\beta \Delta H} \rangle_\Gamma /
\langle e{-\beta \Delta H} \rangle_\Gamma ,
\end{equation}
for all finite $A \subset \Z^d$, where
$\Delta H$ is the (finite) sum
\begin{equation}\label{eight3}
\Delta H = -\sum_{\{x,y\} \in \E^d}
\Delta J_{\{x,y\}} s_x s_y .
\end{equation}
The covariance property that a metastate
$\kappa_{\cal J}$ satisfies is that under all
such finite changes $\Delta {\cal J}$, the
measure $\kappa_{{\cal J}+ \Delta{\cal J}}$ on
states for ${\cal J}+ \Delta{\cal J}$ is the one
induced by $\kappa_{\cal J}$ under the mapping
$\Gamma \to \phi_{\Delta {\cal J}}(\Gamma)$.
That is, changing from $\kappa_{\cal J}$ to
$\kappa_{{\cal J} +\Delta{\cal J}}$
corresponds to simply a change of variables
from $\Gamma$ to $\Gamma' = \phi_{\Delta {\cal J}}
(\Gamma)$.
Although this property is not particularly surprising,
given the nature of the construction of the metastate,
we believe that it is important, both
conceptually and for its possible relevance
to such issues as whether the nonstandard SK picture
is valid. First we discuss a general conceptual point.
Previously, when motivating the construction of metastates
by analogy with dynamical systems, we did not give any
analogue for the dynamical system probability measure
having the property of being invariant under the
dynamics. Now, we propose that the Aizenman-Wehr
covariance property under coupling changes is the
appropriate analogue. Our reasoning is as follows.
Suppose we consider free b.c.'s. Changing from a cube of
size $L$ to one of size $L+1$ corresponds to taking
a certain layer of couplings and changing them from
zero to nonzero values. Having already made an analogy
between $L$ and the time $t$ for the dynamical
system, it seems appropriate to extend it to one
between dynamical invariance ($t \to t+1$) and
coupling covariance (${\cal J} \to {\cal J}+ \Delta {\cal J}$).
The analogy is even clearer if we consider increasing
volumes not by a whole layer at a time but by
a single site at a time.
To explore the possible implications of this covariance
property of the metastate to the nonstandard
SK picture, we will focus on the distribution
of the overlap distribution $P_{\Gamma}$ as
$\Gamma$ varies according to the metastate, when
${\cal J} \to {\cal J}+\Delta{\cal J}$. First,
we remark that it is not hard to see that for any pure
state $P_{\cal J}^\alpha$ for ${\cal J}$,
$\phi_{\Delta {\cal J}} (P_{\cal J}^\alpha)
\equiv P_{{\cal J}+\Delta {\cal J}}^\alpha$
is a pure state for
${\cal J}+ \Delta {\cal J}$. Next we note that if
$\Gamma$ is a countable mixture of pure states,
then it transforms to $\Gamma'$ given by
\begin{equation}\label{eight4}
\phi_{\Delta {\cal J}} (\sum_{\alpha}
W_{\Gamma}^{\alpha} P_{\cal J}^{\alpha}) =
\sum_\alpha W_{\Gamma'}^\alpha P_{{\cal J}+\Delta {\cal J}}^
\alpha ,
\end{equation}
where
\begin{equation}\label{eight5}
W_{\Gamma'}^\alpha = W_\Gamma^\alpha
\langle e^{-\beta \Delta H}\rangle_{\cal J}^\alpha /
\sum_\alpha W_\Gamma^\alpha
\langle e^{-\beta \Delta H}\rangle_{\cal J}^\alpha .
\end{equation}
Furthermore, since $P_{\cal J}^\alpha$ and
$P_{{\cal J}+\Delta{\cal J}}^\alpha$ are clearly
equivalent measures (i.e., are supported on the same
sets of spin configurations) it follows that the
$r_{\alpha,\beta}$'s of $\Gamma'$ are the same
as those of $\Gamma$.
In possibilities 2 or 5 of Sec. 6,
$\langle e^{-\beta \Delta H} \rangle _{\cal J}^\alpha$
is the same for each of the two $\alpha$'s and so
the two weights for $\Gamma'$ are both $1/2$ as they were
for $\Gamma$. But in the nonstandard SK picture, there
seems every reason to expect nontrivial dependence
of $\langle e^{-\beta \Delta H} \rangle _{\cal J}^\alpha$
on the many $\alpha$'s appearing for each $\Gamma$. Thus,
under changes of finitely many couplings, each
$P_\Gamma$ would be changed to a $P_{\Gamma'}$ with the
same $r_{\alpha,\beta}$'s but {\it different} weights.
Nevertheless, by the translation invariance/ergodicity
argument mentioned earlier in this section, the
{\it distribution} of the $P_\Gamma$'s (as $\Gamma$
varies over the metastate) in fact does not depend
on ${\cal J}$ and hence is unchanged by
${\cal J} \to {\cal J}+ \Delta{\cal J}$.
Thus the Aizenman-Wehr covariance property under changes
of couplings places a large number of constraints on the
distribution of the $P_\Gamma$'s that can arise in
the nonstandard SK picture. We wonder whether all
these constraints (which do {\it not} arise
either in possibility 2 or in possibility 5) can
actually be satisfied. Clearly, more study of
this issue is needed.
We close by briefly discussing zero temperature metastates
for the EA model. Let us assume that the common
distribution $\mu$ of the individual couplings is
continuous. As mentioned in the discussion of Sec. 5
on the highly disordered spin glass model, the
metastate at zero
temperature (constructed say with
periodic b.c.'s) must be concentrated on $\Gamma$'s
of the form $\delta_s + \delta_{-s}$, where $s$
is an infinite volume ground state. Thus at zero temperature,
the nonstandard SK picture degenerates into
the picture of our possibility 5. But this many
state picture still differs from the scaling
picture since the metastate would be dispersed
over many choices of $\pm s$ and not on a single
pair of ground states. It would be of some
interest to determine which of these pictures occurs.
The answer is not known even for $d=2$.
\section*{Acknowledgments}
The authors thank Aernout van~Enter for many
valuable suggestions for improving the original draft
of this paper. They also
wish to thank Anton Bovier and Pierre Picco for encouraging the
writing of the paper, for editing the volume in which it
appears and for organizing
the associated conference. C.~M.~N. thanks Anton Bovier and the
Weierstrass Institute for their hospitality and support
during August, 1996, when part of this paper was written.
%\begin{appendix}
\appendix
\section{Appendix: Disordered Systems and Metastates}
%\markboth{APP. DISORDERED SYSTEMS AND METASTATES}{APP. DISORDERED
%SYSTEMS AND METASTATES}
\renewcommand{\theequation}{{\Alph{section}.\arabic{equation}}}
\setcounter{equation}{0}
An infinite volume Gibbs state for couplings $J=(J_e:e\in \E^d)$ (and
inverse temperature $\beta$, which will henceforth be fixed) is a
probability measure on ${\cal S}=\{-1,+1\}^{\Z^d}$ (with the usual
$\sigma$-field generated by cylinder sets) that satisfies extra
requirements (the DLR equations). Let us denote by ${\cal M}({\cal S})$
the set of all probability measures on ${\cal S}$. In this appendix, we
give some technical lemmas and propositions about ${\cal M}({\cal
S})$-valued random variables $Q$ and their distributions $\kappa$ (on
${\cal M}({\cal S})$) and, for disordered systems, about the joint
distributions $\kappa^\dagger$ of $(J,Q)$. Our primary interest is in
$\kappa^\dagger$'s which are supported on configurations $({\cal J},q)$ such that $q$
is an infinite volume Gibbs state for ${\cal J}$. Then the conditional
distribution of $Q$ given $J$ will be a metastate $\kappa_{\cal J}$ $-$ i.e., a
probability measure on ${\cal M}({\cal S})$ supported on the infinite
volume Gibbs states for ${\cal J}$. We will begin however with more general
$Q$'s, $\kappa$'s and $\kappa^\dagger$'s. It should be noted that the material
presented here (other than Lemma \ref{lemmaB5} and Proposition
\ref{propB6}) is basically contained, implicitly or explicitly, in the
appendix of \cite{AW1990}.
It is convenient to regard a probability measure on ${\cal S}$ as a
point $q$ in the product space $\Omega^1\equiv \R^{\cal A}$, where
\begin{equation}\label{B1}
{\cal A}=\bigcup_{\Lambda \subset \Z^d~finite} \left\{ (\Lambda,s): s \in
{\cal S}_\Lambda = \{-1,+1\}^\Lambda \right\},
\end{equation}
with the coordinate $q_{(\Lambda,s)}$ denoting the probability of the
cylinder set $\{s'\in {\cal S}:s_i'=s_i~\forall i\in \Lambda \}$. It is a
standard fact that a point $q\in \Omega^1$ corresponds in this way to a
probability measure if and only if it satisfies the three properties of
positivity, normalization and consistency; i.e., for all finite subsets
$\Lambda$, $\Lambda'$ of $\Z^d$ with $\Lambda \subset \Lambda'$:
\begin{eqnarray}
\label{B2}
&&\forall s \in {\cal S}_\Lambda,\quad q_{(\Lambda,s)} \ge 0, \\
\label{B3}
&& \sum_{s \in {\cal S}_\Lambda} q_{(\Lambda,s)}=1,\\
\label{B4}
&&\forall s \in {\cal S}_\Lambda,\quad q_{(\Lambda,s)}=\sum_{s' \in {\cal
S}_{\Lambda'}:~s'=s~on~\Lambda} q_{(\Lambda',s')}.
\end{eqnarray}
We will identify such a $q$ with the (unique) corresponding probability
measure on ${\cal S}$, and the set of all $q$'s in $\Omega^1$ satisfying
(\ref{B2})-(\ref{B4}) for all finite $\Lambda'\supset \Lambda$
will be identified with ${\cal M}({\cal S})$. Clearly ${\cal M}({\cal
S})$ (now regarded as a subset of $\Omega^1$) is measurable, i.e., it
belongs to ${\cal F}^1$, the product $\sigma$-field over ${\cal A}$ of
the Borel $\sigma$-field ${\cal B}$ on each $\R$.
Our first lemma uses elementary compactness arguments to construct, via
an infinite volume limit, a joint distribution $\kappa^\dagger$ (for some
random $(J,Q)$) on the product space
\begin{equation}\label{B5}
(\Omega^\dagger,{\cal F}^\dagger)=(\Omega^0,{\cal F}^0)\times(\Omega^1,{\cal F}^1),
\end{equation}
where $(\Omega^0,{\cal F}^0)$ is the product over $\E^d$ of $(\R,{\cal
B})$, such that $\kappa^\dagger$ is supported on $\{({\cal J},q):q\in{\cal M}({\cal
S})\}$. For a sequence of subsets $B_n$ of $\E^d$ we write $B_n \to \E^d$
if for every finite $B\subset \E^d$, $B_n\supset B$ for all large $n$;
for subsets $C_n$ of $\Z^d$ we similarly define $C_n\to \Z^d$.
\begin{lemma}\label{lemmaB1} Let $B_n\subset\E^d$ and $C_n\subset\Z^d$
be finite and such that $B_n \to \E^d$ and $C_n \to \Z^d$. Suppose that
for each $n$ there are families of real valued random variables (on some
$(\Omega,{\cal F},\nu)$), $J^n=(J_e^n: e \in B_n)$ and
$Q^n=(Q_{(\Lambda,s)}^n:\Lambda \subseteq C_n,~s\in {\cal S}_\Lambda)$
such that
\begin{equation}\label{B6}
\forall \mbox{ finite }\Lambda'\supset \Lambda,~\forall \mbox{ large
}n,~ q=Q^n\mbox{
satisfies (\ref{B2})-(\ref{B4})}
\end{equation}
and
\begin{equation}\label{B7}
\forall e, \mbox{ the family of distributions of }J_e^n\mbox{ (as $n$
varies) is tight}.
\end{equation}
Then the finite dimensional distributions of $(J^n,Q^n)$ converge along
a subsequence to a probability measure $\kappa^\dagger$ on $(\Omega^\dagger,{\cal
F}^\dagger)$ with $\kappa^\dagger(\Omega^0\times {\cal M}({\cal S}))=1$; i.e., there
exists a subsequence $n_k$ of the $n$'s such that for any $m$, $m' \in
\N$, any $e_1,\dots,e_m\in \E^d$, any
$(\Lambda^1,s_1),\dots,$ $(\Lambda^{m'},s_{m'})\in {\cal A}$ and any
bounded continuous real valued function $f$ on $\R^{m+m'}$
\begin{eqnarray}\label{B8}
E\left(f\left(J_{e_1}^{n_k},\dots,J _{e_m}^{n_k},
Q_{(\Lambda^1,s_1)}^{n_k},\dots,
Q_{(\Lambda^{m'},s_{m'})}^{n_k}\right)\right) \quad \quad \quad\quad \quad \quad\\
\quad \quad \quad\quad \quad \quad \nonumber \to \int_{\Omega^\dagger}
f\left({\cal J}_{e_1},\dots,
{\cal J}_{e_m}, q_{(\Lambda^1,s_1)},\dots, q_{(\Lambda^{m'},s_{m'})} \right)
d\kappa^\dagger(({\cal J},q)).
\end{eqnarray}
Here, $E$ denotes expectation with respect to $\nu$.
\end{lemma}
{\it Proof.} For each $L$, and all sufficiently large $n$, let
$\kappa_{L,n}^\dagger$ denote the joint distribution of the finitely many
variables $J_{\{x,y\}}^n$, with $x$ and $y$ in the cube $\Lambda_L$, and
$Q_{(\Lambda,s)}^n$ with $\Lambda \subseteq \Lambda_L$. The assumption
(\ref{B6}) implies that the one-dimensional marginal
distribution of each
$Q_{(\Lambda,s)}^n$ is supported on $[0,1]$ and so the family of
marginal distributions, as $n$
varies for a fixed $(\Lambda,s)$, is tight; on the
other hand, for a fixed $e$, the tightness of the $J_e^n$ marginal
distributions was
assumed. Thus as $n$ varies for fixed $L$, the $\kappa_{L,n}^\dagger$'s are
tight. It follows that we may choose a subsequence of the $n$'s for each
$L$ (with the one for $L''>L'$ a sub-subsequence of the one for $L'$) along
which $\kappa_{L,n}^\dagger$ converges to some probability measure
$\kappa_{L}^\dagger$. By diagonalization, there is a single subsequence $n_k$
so that for every $L$, $\kappa_{L,n_k}^\dagger\to \kappa_{L}^\dagger$ as $k\to
\infty$.
Now for $L''>L'$, the marginal distribution
of $\kappa_{L'',n_k}^\dagger$ for the variables
coming from $\Lambda_{L'}$ (i.e., $J_{\{x,y\}}^{n_k}$ and
$Q_{(\Lambda,s)}^{n_k}$ with $x,y\in \Lambda_{L'}$ and $\Lambda
\subseteq \Lambda_{L'}$) is just $\kappa_{L',n_k}^\dagger$ and the same is
true for $\kappa_{L''}^\dagger$ and $\kappa_{L'}^\dagger$. Thus the set of
$\kappa_{L}^\dagger$'s is a consistent family of finite dimensional
distributions corresponding to some $\kappa^\dagger$ on $(\Omega^\dagger,{\cal
F}^\dagger)$. This proves (\ref{B8}).
It remains only to show that $\kappa^\dagger(\Omega^0\times {\cal M}({\cal
S}))=1$. But for any fixed finite $\Lambda'\supset \Lambda$, the set of
$q$ in $\Omega^1$ satisfying (\ref{B2})-(\ref{B4}) is a finite cylinder
set corresponding to a {\it closed} subset $F_{\Lambda',\Lambda}$ of
the finitely many variables appearing in (\ref{B2})-(\ref{B4}). By
assumption (\ref{B6}), this subset has $\kappa_{L,n}^\dagger$-probability one
for all large $L$ and $n$, and so, by the already proved convergence in
distribution, the corresponding cylinder set has $\kappa^\dagger$-probability
one. Since this is true for every finite $\Lambda'\supset\Lambda$, it
follows that $\Omega^0\times{\cal M}({\cal S})$ has
$\kappa^\dagger$-probability one, as desired.
\EndProof
In Section 4, we use the following special corollary of Lemma
\ref{lemmaB1}, where $J_e^n=J_e$, with no dependence on $n$, and where
$Q^n$ is a functional of these fixed couplings. There $Q^n$ is given by
a Gibbs distribution on $C_n$ for the fixed couplings.
\begin{lemma}\label{lemmaB2}
Let $B_n$ and $C_n$ be as in Lemma \ref{lemmaB1} and let $(J_e:e\in
\E^d)$ be a fixed family of random variables (on some $(\Omega, {\cal
F},\nu)$). Suppose for each $n$ and each ${\cal J} \in \R^{B_n}$, $P_{\cal J}^n$ is a
probability measure on ${\cal S}_{C_n}$ which depends measurably on ${\cal J}$,
i.e., for each $s' \in {\cal S}_{C_n}$, $P_{\cal J}^n(\{s'\})$, as a
function of
${\cal J} \in \R^{B_n}$, is Borel measurable. For $\Lambda \subseteq C_n$ and
$s \in {\cal S}_\Lambda$, define random variables,
\begin{equation}\label{B9}
Q_{(\Lambda,s)}^n=P_{(J_e:e\in B_n)}^n(\{t\in {\cal S}_{C_n}:t=s \mbox{
on } \Lambda \});
\end{equation}
then along some subsequence $n_k$ of the $n$'s,
\begin{equation}\label{B10}
(J,Q^{n_k})=(J_e,Q_{(\Lambda,s)}^{n_k}:e\in \E^d,~\Lambda \subseteq
C_{n_k}~, s \in {\cal S}_\Lambda) \to \kappa^\dagger
\end{equation}
in the sense of (\ref{B8}), where $\kappa^\dagger$ is some probability measure
on $\Omega^0\times{\cal M}({\cal S})$ (i.e., $\kappa^\dagger$ is a probability
measure on $(\Omega^\dagger,{\cal F}^\dagger)$ with $\kappa^\dagger(\Omega^0\times {\cal
P}({\cal S}) )=1$).
\end{lemma}
{\it Proof.} This is an immediate corollary of Lemma \ref{lemmaB1} by
defining, for each $n$ and each $e\in B_n$, $J_e^n=J_e$.
\EndProof
When $B_n$ contains all $e=\{x,y\}\in \E^d$ with $x,y\in \Lambda_n$ and
$P_{\cal J}^n$ is some Gibbs distribution on ${\cal S}_{C_n}$ (for the
couplings ${\cal J}_e$ and the given $\beta$), then the probabilities
$q_{(\Lambda,s)}^n \equiv P_{\cal J}^n(t=s\mbox{ on } \Lambda)$ satisfy the
following finite volume version of the DLR equations: For $\Lambda,
\Lambda'\subseteq C_n$ with $(\Lambda \cup \partial \Lambda)\subseteq
\Lambda'$,
\begin{equation}\label{B11}
\forall s' \in {\cal S}_{\Lambda
'},\quad q_{(\Lambda',s')}^n=q_{(\Lambda'\setminus \Lambda,
s'|_{\Lambda'\setminus \Lambda})}^n~ P_{\Lambda,\beta}^{s'|_{\partial
\Lambda}}(\{s'|_{\Lambda}\}),
\end{equation}
where $s'|_{A}$ denotes the restriction of $s'$ to $A\subseteq \Lambda'$
and $P_{\Lambda,\beta}^{\bar{s}}$ denotes the usual finite volume Gibbs
distribution on $\Lambda$ for the couplings ${\cal J}_e$, with b.c. $\bar{s}$
on $\partial \Lambda$.
\begin{lemma}\label{lemmaB3}
Suppose that the hypotheses of Lemma \ref{lemmaB2} are valid and, in
addition, the probabilities $q_{(\Lambda,s)}^n \equiv P_{\cal J}^n(t=s\mbox{ on
} \Lambda)$, satisfy (\ref{B11}) for all finite $\Lambda'\supseteq
\Lambda \cup \partial \Lambda$ and all large $n$. Then the $\kappa^\dagger$ of
Lemma \ref{lemmaB2} has
\begin{equation}
\label{B12}
\kappa^\dagger\left( \{ ({\cal J},q):q \mbox{ is an infinite volume Gibbs
distribution for }{\cal J} \} \right)=1.
\end{equation}
\end{lemma}
{\it Proof.} For a fixed $\Lambda'$, $\Lambda$ and $s'$, the condition
(\ref{B11}) is of the form $h=0$ where $h$ is a continuous function of
$q_{(\Lambda',s')}^n$, $q_{(\Lambda'',s'')}^n$ (with
$\Lambda''=\Lambda'\setminus \Lambda$ and
$s''=s'|_{\Lambda'\setminus\Lambda}$) and $J_{e_1},\dots, J_{e_m}$ for
certain fixed edges $e_1,\dots,e_m$. Thus the set of these finitely many
variables satisfying (\ref{B11}) is a closed subset
$F_{\Lambda',\Lambda,s'}$. It follows from convergence in distribution
(as in the proof of Lemma \ref{lemmaB1}) that the corresponding cylinder
set has $\kappa^\dagger$-probability one. Thus the set $H$ of $({\cal J},q)$
such that
$q\in {\cal M}({\cal S})$ and (\ref{B11}) (with $q^n$ replaced by $q$)
is valid for all finite $\Lambda' \supseteq \Lambda \cup \partial
\Lambda$ has $\kappa^\dagger(H)=1$.
We consider a pair $({\cal J},P)\in H$. For a finite $\Lambda$ and for $L$
large enough so that the cube $\Lambda_L$ contains $\Lambda \cup
\partial \Lambda$, we denote by ${\cal F}_{\Lambda_L\setminus \Lambda}$
the $\sigma$-field generated by $\{s_x:x\in \Lambda_L\setminus
\Lambda\}$. The conditions (\ref{B11}) imply that the conditional
probability of $P$ with respect to ${\cal F}_{\Lambda_L\setminus
\Lambda}$ satisfies, for any $t \in {\cal S}_\Lambda$,
\begin{equation}\label{B13}
P\left( \{s\in {\cal S}:s=t \mbox{ on }\Lambda \} | {\cal
F}_{\Lambda_L\setminus \Lambda} \right)= P_{\Lambda,\beta}^{\bar{s}}(t),
\end{equation}
where $\bar{s}\in {\cal S}_{\partial \Lambda}$ is given by the
(conditioned) values of $s_x$ for $x\in \partial \Lambda$. By the
martingale convergence theorem, we may let $L\to \infty$ to get
\begin{equation}\label{B14}
P\left( \{s\in {\cal S}:s=t \mbox{ on }\Lambda \} | {\cal
F}_{\Z^d\setminus \Lambda} \right)= P_{\Lambda,\beta}^{\bar{s}}(t).
\end{equation}
But the set of these equations for arbitrary finite $\Lambda$ and $t \in
{\cal S}_\Lambda$ is just the set of DLR equations which characterize $P$
as an infinite volume Gibbs distribution for the couplings $({\cal J}_e:e\in
\E^d)$. Since the DLR equations also imply the validity of all the
conditions (\ref{B11}), we see that the set of $({\cal J},q)$ appearing in
(\ref{B12}) is identical to $H$. Since $H\in {\cal F}^\dagger$ with
$\kappa^\dagger(H)=1$, the proof is complete.
\EndProof
For any probability measure $\kappa^\dagger$ on $(\Omega^\dagger,{\cal
F}^\dagger)=(\Omega^0,{\cal F}^0)\times(\Omega^1,{\cal F}^1)$, one can define
the marginal distribution (of ${\cal J}$), $\bar{\nu}$, on
$(\Omega^0,{\cal F}^0)$ and the conditional distribution (of $q$, given
${\cal J}$), $\kappa_{\cal J}$, which for $\bar{\nu}$-a.e. ${\cal J}$ is
a probability measure
on $(\Omega^1,{\cal F}^1)$. For any $F \in {\cal F}^\dagger$,
\begin{equation}\label{B15}
\kappa^\dagger(F)=\bar{E}[\kappa_{\cal J}(F_{\cal J})],
\end{equation}
where $\bar{E}$ denotes expectation with respect to $\bar{\nu}$ and
\begin{equation}\label{B16}
F_{\cal J}=\{q\in \Omega^1:({\cal J},q)\in F\}.
\end{equation}
As a corollary of Lemma \ref{lemmaB3}, we have the following.
\begin{lemma}\label{lemmaB4}
Let $\kappa_{\cal J}$ be the conditional distribution of $q$, given
${\cal J}$, for
the $\kappa^\dagger$ of Lemma \ref{lemmaB2}. Under the hypotheses of Lemma
\ref{lemmaB3},
$\mbox{for $\bar{\nu}$-a.e. }{\cal J}, \hfill $
$\kappa_{\cal
J}\left(\{q:q\mbox{ is an infinite
volume Gibbs distribution for }{\cal J}\}\right)=1.$
\end{lemma}
{\it Proof.} This is an immediate consequence of (\ref{B15}) by taking
$F$ to be the event appearing in (\ref{B12}).
\EndProof
The rest of the appendix is concerned, in the context of Lemma
\ref{lemmaB2}, with the convergence of empirical distributions to
$\kappa_{\cal J}$.
For each $(\Lambda,s)\in {\cal A}$, $Q_{(\Lambda,s)}^n$ is a random
variable, defined for $\{n:\Lambda\subseteq C_n\}$ and thus for all
large $n$; we make the convention that $Q_{(\Lambda,s)}^n=0$ for those
finitely many $n$'s for which $\Lambda \not\subseteq C_n$. We consider,
for each $L$, the finite dimensional random vector
\begin{equation}\label{B18}
\vec{Q}_L^n=\left(Q_{(\Lambda,s)}^n:\Lambda\subseteq \Lambda_L,~s\in
{\cal S}_\Lambda \right),
\end{equation}
and the empirical distributions, along some subsequence $n_k'$,
\begin{equation}\label{B19}
{\kappa}^{L,K}=\frac{1}{K}~\sum_{k=1}^K \delta_{\vec{Q}_L^{n_k'}}.
\end{equation}
These are (random) probability measures on $\R^{m_L}$, for some finite
$m_L$.
In the context of Lemma \ref{lemmaB2}, ${\kappa}^{L,K}$ is random
because it depends (measurably) on $J$. To denote this dependence, we will
sometimes write ${\kappa}^{L,K}_J$. We are interested in whether and in
what sense ${\kappa}^{L,K}$ converges as $K\to \infty$ to some limit
(random) probability measure on $\R^{m_L}$. We note that there is a
natural candidate for the limit measure $-$ namely, take the conditional
distribution $\kappa_{\cal J}$, of $q$, given ${\cal J}$, for the joint
distribution $\kappa^\dagger$ and consider its marginal distribution
$\kappa_{\cal J}^L$ for $\left(
q_{(\Lambda,s)}:\Lambda\subseteq\Lambda_L,~s\in {\cal S}_\Lambda
\right)$. The next lemma is a technical result which leads to the
subsequent proposition giving a.s. convergence to this limit along a
subsequence.
\begin{lemma}\label{lemmaB5}
Assume the hypotheses of Lemma \ref{lemmaB2}. For any fixed $L$ and
fixed $\vec{r}\in \R^{m_L}$, there exists a sub-subsequence $n_k'$ of
the subsequence $n_k$ of Lemma \ref{lemmaB2}, such that (with
${\kappa}^{L,K}$ given by (\ref{B18})-(\ref{B19})), the sequence of
random
variables,
\begin{equation}\label{B20}
\Phi_{L,\vec{r}}^K=\int\limits_{\R^{m_L}}
e^{i~(\vec{r},\vec{q}_L)}d{\kappa}^{L,K}
(\vec{q}_L)=\frac{1}{K}~\sum_{k=1}^K e^{i~(\vec{r},
\vec{Q}_L^{n_k'})},
\end{equation}
(where $(\vec{r},\vec{q}_L)$ denotes the standard inner product on
$\R^{m_L}$) converges in \linebreak $L^2(\Omega,{\cal F},\nu)$, as $K\to
\infty$,
to some random variable $\Phi_{L,\vec{r}}$.
\end{lemma}
{\it Proof.} Let $Y_n=e^{i~(\vec{r}, \vec{Q}_L^{n})}$. We must show
that $n_k'$ can be chosen so that
\begin{eqnarray}
E\left(\Phi_{L,\vec{r}}^{K'}-\Phi_{L,\vec{r}}^K\right)^2&=&
\frac{1}{{K'}^2}\sum_{k,j=1}^{K'} E\left(Y_{n_k'}Y_{n_j'}\right) +
\frac{1}{K^2}\sum_{k,j=1}^K E\left(Y_{n_k'}Y_{n_j'}\right) \nonumber
\\
& & \quad \label{B21}
-\frac{2}{K'K}\sum_{k=1}^{K'}\sum_{j=1}^K
E\left(Y_{n_k'}Y_{n_j'}\right) \to 0
\end{eqnarray}
as $K,K'\to \infty$. Note that the total contribution of the diagonal
terms (where $k=j$) in the last expression is negligible as $K,K'\to
\infty$, as is the contribution from terms with $k$ (or $j$) fixed. It
follows that it suffices to choose $n_k'$ so that
\begin{equation}\label{B22}
\lim_{\stackrel{k,j\to \infty}{k\neq j}}
E\left(Y_{n_k'}Y_{n_j'}\right) \mbox{ exists.}
\end{equation}
We will do this by a compactness argument.
We will use the facts that $C_{mn}\equiv E(Y_mY_n)$ is bounded in
modulus by $1$ and that $C_{mn}=C_{nm}$. We begin by choosing, for each
$l$, a subsequence of the $n$'s (with the one for $l''>l'$ a subsequence
of the one for $l'$) along which $C_{ln}$ converges to some
$\alpha_l$. By diagonalization, there is a fixed subsequence
$\tilde{n}_k$ so that, for every $l$, $C_{l\tilde{n}_k}\to \alpha_l$ as
$k\to \infty$. Since the $\alpha_l$'s are also bounded in modulus by
one, we may next pick a sub-subsequence $n_k^\ast$ of $\tilde{n}_k$ so
that $\alpha_{n_k^\ast}\to \alpha$. We now have:
\begin{equation}\label{B23}
\forall k,\quad C_{n_k^\ast,n_{k'}^\ast} \to \alpha_{n^\ast_k} \quad \mbox{ as }
k'\to \infty, \quad \mbox{ and }\quad \alpha_{n^\ast_k}\to \alpha \quad \mbox{ as }k\to \infty.
\end{equation}
Finally, we define $n_k'$ inductively by taking $n_1'=n_1^\ast$
($=n_{k_1}^\ast$ with $k_1=1$) and, for $j\ge 1$, taking
$n_{j+1}'=n_{k_{j+1}}^\ast$ with $k_{j+1}$ the smallest $k'>k_j$ such
that
\begin{equation}\label{B24}
\left|C_{n_j'n_k^\ast}-\alpha_{n_j'}\right| < \frac{1}{2^j} \quad \mbox{ for
}k\ge k'.
\end{equation}
It follows that
\begin{equation}\label{B25}
\lim_{j\to\infty}\sup_{k>j}\left| C_{n_j'n_k'} -\alpha \right|
\le \lim_{j\to\infty}\left(\frac{1}{2^j}+\left|\alpha_{n_j'}-\alpha
\right| \right) =0.
\end{equation}
But this, together with $C_{mn}=C_{nm}$, implies (\ref{B22}).
\EndProof
\begin{prop}\label{propB6}
Assume the hypotheses of Lemma \ref{lemmaB2}. Then there exists a
sub-subsequence $n_k'$ of the subsequence $n_k$ of Lemma \ref{lemmaB2}
and some subsequence $K_m$ of the $K$'s, such that, almost surely, the
empirical distributions ${\kappa}^{L,K_m}$ (given by
(\ref{B18})-(\ref{B19})) converge (weakly) as $m \to
\infty$ to the finite dimensional marginal
distributions $\kappa_J^L$ (of the
conditional distribution $\kappa_{\cal J}$ of the $\kappa^\dagger$ of Lemma
\ref{lemmaB2}, with ${\cal J}=J$).
I.e., for $\nu$-a.e. $\omega$, $\forall L$, $\forall$ bounded continuous
functions $f$ on $\R^{m_L}$,
\begin{equation}\label{B26}
\lim_{m\to \infty} \int\limits_{\R^{m_L}}
f(\vec{q}_L)d{\kappa}^{L,K_m}_{J(\omega)}(\vec{q}_L) = \int_{\Omega^1}
f\left( (q_{(\Lambda,s)}:\Lambda \subseteq \Lambda_L,~s \in {\cal
S}_\Lambda)\right) d\kappa_{J(\omega)}(q).
\end{equation}
\end{prop}
{\it Proof.} By Lemma \ref{lemmaB5} and a diagonalization argument, we
can replace the $n_k'$ of Lemma \ref{lemmaB5} by a single further
sub-subsequence (which we will still call $n_k'$) so that
$\Phi_{L,\vec{r}}^K$ converges in $L^2$ to some $\Phi_{L,\vec{r}}$, as
$K\to \infty$, for every $L$ and every $\vec{r}\in \R^{m_L}$ with
rational coordinates. But $L^2$ convergence implies convergence in
probability, which implies a.s. convergence along a subsequence of
$K$'s. Thus, by a further diagonalization, we have for a single
subsequence $K_m$ of the $K$'s that $\Phi_{L,\vec{r}}^{K_m}$ converges
a.s. to $\Phi_{L,\vec{r}}$, as $m\to \infty$, for all $L$ and rational
$\vec{r}$.
Now for each $\omega$, $\Phi_{L,\vec{r}}^K$, as a function of $\vec{r}$,
is the characteristic function of a probability measure
${\kappa}_{J(\omega)}^{L,K}$ supported on the compact set $[0,1]^{m_L}$
(since each $Q_{(\Lambda,s)}^n$ takes values in $[0,1]$). The tightness
of this family of measures (as $K$ varies) along with convergence (for
$K=K_m$) of the characteristic functions for a dense set of $\vec{r}~$'s
implies that for $\nu$-a.e. $\omega$,
\begin{equation}\label{B27}
\forall L,~\forall \vec{r}\in \R^{m_L}, \quad \Phi_{L,\vec{r}}^{K_m}\to
\Phi_{L,\vec{r}}= \int\limits_{\R^{m_L}}
e^{i~(\vec{r},\vec{q}_L)}d\kappa^L(\vec{q}_L)\quad \mbox{ as }m\to \infty,
\end{equation}
where $\kappa^L$ is some probability measure on $\R^{m_L}$, depending on
$\omega$.
To complete the proof of the proposition, it only remains to identify
$\kappa^L$ with the marginal distribution of
$\kappa_{J(\omega)}$. Because of the relation between $\kappa_{\cal J}$
and $\kappa^\dagger$, to do this, it suffices to show that for every bounded
continuous function $g({\cal J})$, depending on only finitely many
${\cal J}_e$'s,
\begin{equation}\label{B28}
E\left(g(J)\Phi_{L,\vec{r}}\right)= \int g({\cal
J})e^{i~(\vec{r},\vec{q}_L)}d\kappa^\dagger(({\cal J},q)).
\end{equation}
But by (\ref{B20}) and Lemma \ref{lemmaB2}
\begin{eqnarray}\nonumber
E\left(g(J)\Phi_{L,\vec{r}}\right)=
\lim_{m\to\infty}E\left(g(J)\Phi_{L,\vec{r}}^{K_m}\right)
&&\hspace*{-2ex}=\lim_{m\to\infty}\frac{1}{K_m}~\sum_{k=1}^{K_m} E\left(g(J)
e^{i~(\vec{r}, \vec{Q}_L^{n_k'})}\right)\\
&&\hspace*{-2ex}= \int_{\Omega^\dagger} g({\cal J})
e^{i~(\vec{r},\vec{q}_L)}d\kappa^\dagger(({\cal J},q)) ,\nonumber\\
\label{B29}
\end{eqnarray}
as desired.
\EndProof
%\end{appendix}
\bibliographystyle{alpha}
\begin{thebibliography}{4cm}
%\markboth{BIBLIOGRAPHY}{BIBLIOGRAPHY}
\bibitem[A80]{A1980}M. Aizenman, Translation invariance and
instability of phase coexistence in the two dimensional Ising System,
Commun. Math. Phys. 73, 1980, pp. 83-94.
%\bibitem[ACCN87]{ACCN1987}M. Aizenman, J. Chayes, L. Chayes,
%C.M. Newman, The phase boundary in dilute and random Ising and Potts
%ferromagnets, J. of Physics A: Mathematical and General 20, 1987,
%pp. 313-318.
%\bibitem[ACCN88]{ACCN1988}M. Aizenman, J. Chayes, L. Chayes,
%C.M. Newman, Discontinuity of the magnetization in one-dimensional
%$1/|x-y|^2$ Ising and Potts models, J. of Stat. Phys. 50, 1988,
%pp. 1-40.
%\bibitem[AKN87]{AKN1987}M. Aizenman, H. Kesten and C.M. Newman,
%Uniqueness of the infinite cluster and continuity of connectivity
%functions for short and long range percolation,
%Commun. Math. Phys. 111, 1987, pp. 505-531.
\bibitem[AW90]{AW1990}M. Aizenman and J. Wehr, Rounding effects of
quenched randomness of first-order phase transitions,
Commun. Math. Phys. 130, 1990, pp. 489-528.
%\bibitem[AlS92]{AS1992}D. Aldous and J.M. Steele, Asymptotics for
%Euclidean minimal spanning trees on random points, Prob. Th. Rel. Fields
%92, 1992, pp. 247-258.
%\bibitem[Ale95]{A1995}K. Alexander, Percolation and minimal spanning
%forests in infinite graphs, Ann. Prob. 23, 1995, pp. 87-104.
\bibitem[AmPZ92]{AmPZ1992}J.M.G. Amaro de Matos, A.E. Patrick and
V.A. Zagrebnov, Random infinite-volume Gibbs states for the Curie-Weiss
random field Ising model, J. Stat. Phys. 66, 1992, pp. 139-164.
%\bibitem[BD86]{BD1986}L.A. Bassalygo and R.L. Dobrushin, Uniqueness
%of a Gibbs field with random potential - an elementary approach,
%Theory Prob. Appl. 31, 1986, pp. 572-589.
%\bibitem[BM94]{BM1994}J. van den Berg and C. Maes, Disagreement
%percolation in the study of of Markov fields, Ann. Prob. 22, No. 2,
%1994, pp. 749-763.
\bibitem[B92]{B1992}J.P.~Bouchaud, Weak ergodicity-breaking and aging
in disordered systems, J.~Phys.~I~(France) 2, 1992, pp. 1705-1713.
\bibitem[BF86]{BF1986}A. Bovier and J. Fr\"ohlich, A heuristic theory
of the spin glass phase, J. Stat. Phys. 44,
1986, pp. 347-391.
%\bibitem[BoP97]{BoP1997}A. Bovier and P. Picco (eds.), Mathematics of
%Spin Glasses and Neural Networks, Birkh\"auser, Boston, 1997 (to appear).
\bibitem[BM87]{BM1987}A.J. Bray and M.A. Moore, Chaotic nature of the
spin-glass phase, Phys. Rev. Lett. 58,
1987, pp. 57-60.
\bibitem[BrK88]{BriK1988}J. Bricmont and A. Kupiainen, Phase
transition in the $3d$ random field Ising model, Commun. Math. Phys. 116,
1988, pp. 539-572.
%\bibitem[BroH57]{BH1957}S.R. Broadbent and J.M. Hammersley, Percolation
%processes I. Crystals and mazes, Proc. Camb. Phil. Soc. 53, 1957,
%pp. 629-641.
%\bibitem[BuK89]{BK1989}R. Burton and M. Keane, Density and uniqueness
%in percolation, Commun. Math. Phys. 121, 1989, pp. 501-505.
%\bibitem[CKLW82]{CKLW1982}R. Chandler, J. Koplik, K. Lerman and
%J.F. Willemsen, Capillary displacement and percolation in porous media,
%J. Fluid Mech. 119, 1982, pp. 249-267.
\bibitem[COE87]{COE1987}M. Campanino, E. Olivieri and A.C.D. van
Enter, One dimensional spin glasses with potential decay
$1/r^{1+\varepsilon}$. Absence of phase transitions and
cluster properties, Commun. Math. Phys. 108, 1987, pp. 241-255.
\bibitem[ChCN85]{CCN1985}J. Chayes, L. Chayes and C.M. Newman, The
stochastic geometry of invasion percolation,
Commun. Math. Phys. 101, 1985, pp. 383-407.
\bibitem[CiMB94]{CiMB1994}M. Cieplak, A Maritan and
J.R. Banavar, Optimal paths and domain walls in
the strong disorder limit, Phys. Rev. Lett. 72,
1994, pp. 2320-2323.
\bibitem[Co89]{Co1989}F. Comets, Large deviation estimates for
a conditional probability distribution. Applications to random
interaction Gibbs measures, Prob. Theory Rel. Fields 80,
1989, pp. 407-432.
%\bibitem[D68]{D1968}R.L. Dobrushin, The description of a random field
%by means of conditional probabilities and conditions of its regularity,
%Theory of Prob. and its Appl. 13, 1968, pp. 197-224.
%\bibitem[DS85]{DS1985}R.L. Dobrushin and S.B. Shlosman, Constructive
%criteria for the uniqueness of a Gibbs field, pp. 371-403, in
%Statistical Mechanics and Dynamical Systems, J. Fritz, A. Jaffe and
%D. Sz\'asz (eds.), Birkh\"auser, Boston, 1985.
%\bibitem[DrKP95]{DKP1995}H. von Dreifus, A. Klein and J. Fernando
%Perez, Taming Griffiths' singularities: infinite differentiability
%of quenched correlation functions, Commun. Math. Phys. 170, 1995,
%pp. 21-39.
\bibitem[DT85]{DT1985}B.~Derrida and G.~Toulouse, J.~Phys.~Lett.~(Paris),
Sample to sample fluctuations in the random energy
model, 46, 1985, pp. L223-L228.
\bibitem[EA75]{EA1975}S.F. Edwards and P.W. Anderson, Theory of spin
glasses, J. Phys. F 5, 1975, pp. 965-974.
%\bibitem[ES88]{ES1988}R.G. Edwards and A.D. Sokal, Generalization of
%the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo
%algorithm, The Physical Review D 38, 1988, pp. 2009-2012.
\bibitem[En90]{E1990}A.C.D. van Enter, Stiffness exponent, number of
pure states, and Almeida-Thouless line in spin-glasses,
J. Stat. Phys. 60, 1990, pp. 275-279.
\bibitem[EnF85]{EnF1985}A.C.D. van Enter and J. Fr\"{o}hlich,
Absence of symmetry breaking for $N$-vector spin glass models
in two dimensions, Commun. Math. Phys. 98, 1985, pp. 425-432.
\bibitem[EnHM92]{EnHM1992}A.C.D. van Enter, A. Hof and
J. Mi\c{e}kisz, Overlap distributions for deterministic
systems with many pure states, J. Phys. A 25, 1992,
pp. L1133-L1137.
\bibitem[FH86]{FH1986}D.S. Fisher and D.A. Huse, Ordered phase of
short-range Ising spin-glasses, Phys. Rev. Lett. 56,
1986, pp. 1601-1604.
\bibitem[FH88]{FH1988}D.S. Fisher and D.A. Huse, Equilibrium behavior
of the spin-glass ordered phase, Phys. Rev. B 38,
1988, pp. 386-411.
\bibitem[FS90]{FS1990}M.E. Fisher and R.R.P. Singh, Critical Points,
large-dimensionality expansions and Ising spin glass, pp. 87-111, in
Disorder in physical systems, G.R. Grimmett and D.J.A. Welsh (eds.),
Clarendon Press, Oxford, 1990.
\bibitem[FSS76]{FSS1976}J. Fr\"{o}hlich, B. Simon and T. Spencer,
Infrared bounds, phase transitions and continuous symmetry
breaking, Commun. Math. Phys. 50, 1976, pp. 79-85.
\bibitem[FZ87]{FZ1987}J. Fr\"{o}hlich and B. Zegarlinski,
The high-temperature phase of long-range spin glasses,
Commun. Math. Phys. 110, 1987, pp. 121-155.
%\bibitem[FoLN91]{FLN1991}G. Forgacs, R. Lipowsky and
%T.M. Nieuwenhuizen, The behaviour of interfaces in ordered and
%disordered systems, pp. 135-363, in Phase Transitions and Critical
%Phenomena, C. Comb and J. Lebowitz (eds.),
%Vol. 14, Acad. Press, London, 1991.
%\bibitem[For72]{F1972}C.M. Fortuin, On the random-cluster model. III. The
%simple random-cluster model, Physica 59, 1972, pp. 545-570.
%\bibitem[ForK72]{FK1972}C.M. Fortuin and P.W. Kasteleyn, On the
%random-cluster model. I. Introduction and relation to other models,
%Physica 57, 1972, pp. 536-564.
%\bibitem[ForKG71]{FKG1971}C.M. Fortuin, P.W. Kasteleyn and J. Ginibre,
%Correlation inequalities on some partially ordered set,
%Commun. Math. Phys. 22, 1971, pp. 89-103.
\bibitem[GKN92]{GKN1992}A. Gandolfi, M. Keane and C.M. Newman,
Uniqueness of the infinite component in a random graph with
applications to percolation and spin glasses, Prob. Theory Rel.
Fields 92, 1992, pp. 511-527.
\bibitem[GNS93]{GNS1993}A. Gandolfi, C.M. Newman and D.L. Stein,
Exotic states in long-range spin glasses, Commun. Math. Phys. 157,
1993, pp. 371-387.
%\bibitem[GS96]{GS1996}A. Gandolfi and E. de Santis, private communication,
%June, 1996.
\bibitem[Ge88]{G1988}H.O. Georgii, Gibbs Measures and Phase Transitions,
de Gruyter Studies in Math., Bd. 9, Berlin, 1988.
%\bibitem[GiM95]{GiM1995}G. Gielis and C. Maes, The uniqueness regime
%of Gibbs fields with unbounded disorder, J. Stat. Phys. 81, 1995,
%pp. 829-835.
%\bibitem[Gr67]{G19}R.B. Griffiths, Correlation in Ising ferromagnets I
%and II, J. Math. Phys. 8, 1967, pp. 478-489.
%\bibitem[Gri94]{G1994}G.R. Grimmett, Percolative problems,
%pp. 69-86, in Probability and Phase
%Transition, G. Grimmett (ed.), Kluwer, Dordrecht, 1994.
\bibitem[Gu95]{G1995}F. Guerra, private communication, September, 1995.
%\bibitem[H57]{H1957}J.M. Hammersley, Percolation processes. Lower
%bounds for the critical probabiliy, Ann. Math. Stat. 28, 1957,
%pp. 790-795.
%\bibitem[H59]{H1959}J.M. Hammersley, Bornes sup\'erieures de la
%probabilit\'e critique dans un processus de filtration, pp. 17-37, in Le
%Calcul des Probabilit\'es et ses Applications, CNRS, Paris, 1959.
%\bibitem[HW65]{HW1965}J.M. Hammersley and D.J.A. Welsh, First-passage
%percolation, subadditive processes, stochastic networks and generalized
%renewal theory, pp. 61-110, in Bernoulli,
%Bayes, Laplace Anniversary Volume, J. Neyman and L. Lecam (eds.),
%Springer-Verlag, Berlin, 1965.
%\bibitem[HaS90]{HS1990}T. Hara and G. Slade, Mean-field critical
%behavior for percolation in high dimensions, Commun. Math. Phys. 128,
%1990, pp. 333-391.
%\bibitem[Har60]{H1960}T.E. Harris, A lower bound for the critical
%probability in a certain percolation process, Proc. Camb. Philosophical
%Soc. 56, 1960, pp. 13-20.
\bibitem[Hi81]{H1981}Y. Higuchi, On the absence of non-translation
invariant Gibbs states for the two-dimensional Ising model, Vol. I,
pp. 517-534,
in Random Fields, Esztergom (Hungary) 1979, J. Fritz, J.L. Lebowitz and
D. Sz\'asz (eds.), Amsterdam, North Holland.
\bibitem[HoJY83]{HJY1983}A. Houghton, S. Jain and A.P. Young,
Role of initial conditions in spin glass dynamics
and significance of Parisi's $q(x)$, J. Phys. C 16, 1983, pp. L375-L381.
%\bibitem[HowN96]{HN1996}C.D. Howard and C.M. Newman, Euclidean models of
%first-passage percolation, preprint, 1996.
\bibitem[HuF87]{HuF1987}D.A. Huse and D.S. Fisher, Pure states in
spin glasses, J. Phys. A 20, 1987, pp. L997-L1003.
%\bibitem[HuH85]{HH}D.A. Huse and C.L. Henley, Pinning and roughening of
%domain walls in Ising systems due to random impurities,
%Phys. Rev. Lett. 54, 1985, pp. 2708-2711.
%\bibitem[HuHF85]{HHF}D.A. Huse, C.L. Henley and D.S. Fisher, Respond,
%Phys. Rev. Lett. 55, 1985, p. 2924.
\bibitem[I85]{I1985}J. Imbrie, The ground state of the
three-dimensional random-field Ising model, Commun. Math. Phys. 98,
1985, pp. 145-176.
%\bibitem[K85]{K}M. Kardar, Roughening by impurities at finite
%temperature, Phys. Rev. Lett. 55, 1985, p. 2923.
%\bibitem[KPZ86]{KP2}M. Kardar, G. Parisi and Y.-C. Zhang, Dynamic scaling
%of growing interfaces, Phys. Rev. Lett. 56, 1986, pp. 889-892.
%\bibitem[KaO88]{KO1988}Y. Kasai and A. Okiji, Percolation problem
%describing $\pm J$ Ising spin glass system, Progress in Theoretical
%Physics 79, 1988, pp. 1080-1094.
%\bibitem[KasF69]{KF1969}P.W. Kasteleyn and C.M. Fortuin, Phase
%transitions in lattice systems with random local properties, Journal of
%the Physical Society of Japan, 26, 1969, pp. 11-14.
%\bibitem[KeS68]{KS19}D.G. Kelly and S. Sherman, General Griffiths'
%inequalities on correlations in Ising ferromagnets, J. Math. Phys. 9, 1968,
%pp. 466-484.
%\bibitem[Kes80]{K1980}H. Kesten, The critical probability of bond
%percolation on the square lattice equals $1/2$, Commun. Math. Phys. 74,
%1980, pp. 41-59.
\bibitem[Ku96]{Ku1996}C. K\"ulske, Metastates in disordered mean
field models: random field and Hopfield models, preprint, 1996.
%\bibitem[LM72]{LM1972}J.L. Lebowitz and A. Martin-L\"of, On the
%uniqueness of the equilibrium state for Ising spin systems, Commun.
%Math. Phys. 25, 1972, pp. 276-282.
\bibitem[L77]{Le1977}F. Ledrappier, Pressure and variational
principle for random Ising model, Commun. Math. Phys. 56,
1977, pp. 297-302.
\bibitem[LY52]{LY1952}T.D. Lee and C.N. Yang, Statistical theory
of equations of state and phase transitions II. Lattice gas
and Ising model, Phys. Rev. 87, 1952, pp. 410-419.
%\bibitem[LeB80]{LB1980}R. Lenormand and S. Bories, Description d'un
%m\'ecanisme de connexion de liaison destin\'e \`a l'\'etude du drainage
%avec pi\`egeage en milieu poreux, C. R. Acad. Sci. Paris S\'er. B 291,
%1980, pp. 279-282.
%\bibitem[LiN96]{LN1996}C. Licea and C.M. Newman, Geodesics in two-dimensional
%first-passage percolation, Ann. Prob. 24, 1996, to appear.
%\bibitem[LiNP96]{LNP1996}C. Licea, C.M. Newman and M.S.T Piza,
%Superdiffusivity in first-passage percolation, Prob. Theory Rel. Fields,
%to appear.
\bibitem[M84]{M1984}W.L. McMillan, Scaling theory of
Ising spin glasses, J. Phys. C 17, 1984, pp. 3179-3187.
\bibitem[MPSTV84]{MPSTV1984}M. M\'ezard, G. Parisi, N. Sourlas,
G. Toulouse and M.A. Virasoro, Nature of spin-glass phase,
Phys. Rev. Lett. 52, 1984, pp. 1156-1159.
\bibitem[MPV85]{MPV1985}M.~M\'ezard, G.~Parisi and M.A.~Virasoro,
Random free energies in spin glasses, J.~Phys.~Lett.~(Paris) 46, 1985,
pp.~L217-L222.
\bibitem[MPV87]{MPV1987}M. M\'ezard, G. Parisi and M.A. Virasoro, Spin
Glass Theory and Beyond, World Scientific, Singapore, 1987.
%\bibitem[N91]{N1991}C.M. Newman, Ising models and dependent
%percolation, pp. 395-401, in Topics in Statistical Dependence, H.W. Block,
%A.R. Sampson and T.H Savits (eds.), IMS Lecture Notes - Monograph Series
%16, 1991.
\bibitem[N94]{N1994}C.M. Newman, Disordered Ising systems and random cluster
representation, pp. 247-260, in Probability and Phase
Transition, G. Grimmett (ed.), Kluwer, Dordrecht, 1994.
%\bibitem[N95]{N1995}C.M. Newman, A surface view of first-passage
%percolation, pp. 1017-1023, in Proceedings of the International Congress
%of Mathematicians, S.D. Chatterji (ed.),
%Birkh\"auser Verlag, Basel, 1995.
\bibitem[N97]{N1997}C.M. Newman, Topics in Disordered Systems,
Birkh\"{a}user, 1997, to appear.
\bibitem[NS92]{NS1992}C.M. Newman and D.L. Stein, Multiple states and
thermodynamic limits in short ranged Ising spin glass models,
Phys. Rev. B 46, 1992, pp. 973-982.
\bibitem[NS94]{NS1994}C.M. Newman and D.L. Stein, Spin glass model
with dimension-dependent ground state multiplicity,
Phys. Rev. Lett. 72, 1994, pp. 2286-2289.
\bibitem[NS96a]{NS1996a}C.M. Newman and D.L. Stein, Ground-state structure in
a highly disordered spin-glass model, J. Stat. Phys. 82,
1996, pp. 1113-1132.
\bibitem[NS96b]{NS1996b}C.M. Newman and D.L. Stein, Non-mean-field
behavior of
realistic spin glass, Phys. Rev. Lett. 76, 1996, pp. 515-518.
\bibitem[NS96c]{NS1995a}C.M. Newman and D.L. Stein, Spatial
inhomogeneity and thermodynamic chaos, Phys. Rev. Lett. 76, 1996,
pp. 4821-4824.
\bibitem[O44]{O1944}L. Onsager, Crystal statistics I. A two-dimensional
model with an order-disorder transition, Phys. Rev. 65, 1944,
pp. 117-149.
\bibitem[P79]{P1979}G. Parisi, Infinite number of order parameters for
spin-glasses, Phys. Rev. Lett. 43, 1979, pp. 1754-1756.
\bibitem[P83]{P1983}G. Parisi, Order parameter for spin-glasses,
Phys. Rev. Lett. 50, 1983, pp. 1946-1948.
\bibitem[PS91]{PS1991}L.A. Pastur and M.V. Shcherbina, Absence of
self-averaging of the order parameter in the
Sherrington-Kirkpatrick Model, J. Stat. Phys. 61, 1991,
pp. 1-19.
%\bibitem[P96]{P1996}A. Pisztora, Surface order large deviations for
%Ising, Potts and percolation models, Prob. Theory Rel. Fields 104,
%1996, pp. 427-466.
%\bibitem[S69]{S1969}T.K. Sarkar, Some lower bound of reliability,
%Technical Report No. 124, Dept. of Operations Research and Statistics,
%Standford University, Stanford, CA, 1969.
\bibitem[S95]{S1995}T. Sepp\"{a}l\"{a}inen, Entropy, limit
theorems and variational principle for disordered lattice systems,
Commun. Math. Phys. 171, 1995, pp. 233-277.
\bibitem[SK75]{SK1975}D. Sherrington and S. Kirkpatrick, Solvable
model of a spin glass,
Phys. Rev. Lett. 35, 1975, pp. 1792-1796.
%\bibitem[SW87]{SW1987}R.H. Swendsen and J.S. Wang, Nonuniversal
%critical dynamics in Monte Carlo simulations, Phys. Rev. Lett. 58, 1987,
%pp. 86-88.
%\bibitem[VW92]{VW1992}M.Q. Vahidi-Asl and J.C. Wierman, A shape
%result for first-passage percolation on the Voronoi tesselation
%and Delaunay triangulation, pp. 247-262, in Random Graphs '89,
%Wiley, New York, 1992.
%\bibitem[WW83]{WW1983}D. Wilkinson and J.F. Willemsen, Invasion
%percolation: A new form of percolation theory, J. Phys. A 16, 1983,
%pp. 3365-3376.
\bibitem[YL52]{YL1952}C.N. Yang and T.D. Lee, Statistical theory
of equations of state and phase transitions I. Theory of
condensation, Phys. Rev. 87, 1952, pp. 404-409.
%\bibitem[Z96]{Z1996}M.P.W. Zerner, private communication, April, 1996.
\end{thebibliography}
%\input{newbook1.ind}
\end{document}