%%%%%%%%%%%%%%%%%%%%%%%%%%
% LATEX DOCUMENT.
%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The main change needed to fit the Journal's style,
% is at the section headers, as explained below.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentstyle[12pt]{article}
%\input{psfig} %preparing for figures
\input epsf %figures program compatible with Apple and Sun
\date {{\small December 10, 1997}} % TRY TO KEEP UPDATED
\bibliographystyle{mabib}
% for BibTeX - sorted numerical labels by order of
% first citation.
\setlength{\oddsidemargin}{.3in}
\setlength{\evensidemargin}{.3in}
\setlength{\textwidth}{6.2in}
\setlength{\textheight}{8.3in}
\setlength{\topmargin}{0in}
\setlength{\leftmargin}{-0.3in}
\parskip=7pt
\parindent 0.4in
%%%%%%%%%%--Macro producing the abstract -- invoked by "\abst{text}:
\def\abst#1{\begin{minipage}{5.25in}
{\noindent \normalsize
{\bf Abstract} #1} \\ \end{minipage} }
%%%%%%%%%% Michael's shortcuts %%%%%%%%%%%%
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\newcommand{\eq}[1]{eq.~(\ref{#1})} %% to invoke write: \eq{...}
\def\proof{ \noindent {\bf Proof:} }
\def\C{{\mathcal C}} %%% to denote connected clusters
\def\E{{\mathrm E}} %%% for "expectation value" (in math mode)
\def\F{{\mathcal F}} %%% to denote connected clusters
\def\I{\mathrm I} %%% \I-- for the indicator function (in math)
\def\P{Prob} %%% appears in many equations Prob_{p_c}
\def\too#1{\parbox[t]{.4in} {$\longrightarrow\\[-9pt] {\scriptstyle #1}$}}
% converges to (?) as #1 to infinity
\def\dim{\overline{dim}_{B}}
\def\liminf{\mathop{\underline{\rm lim}}}
\def\limsup{\mathop{\overline{\rm lim}}} %%%%% prettier limsup, liminf
%\def\Ltoo{\parbox[t]{.4in}
% {$\longrightarrow \\ {\scriptstyle L \to \infty}$}}
% % \Ltoo : ... as L to infinity)
\def\lg{\stackrel{\scriptstyle <}{_{_{\scriptstyle >} }} }
%% \lg for "less and greater"
%%%%%%%%%% Almut's shortcuts %%%%%%%%%%%%
\def\blackbox{{\vrule height 1.3ex width 1.0ex depth -.2ex}\hskip 1.5
truecm}
\def\Chi {{\cal X}}
\def\d {{\rm d}}
\def\eps {\varepsilon}
\def\R {{\bf R}}
\def\abs#1{\left\vert #1 \right\vert}
\def\norm#1{\left\| #1 \right\|}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Explanation:
% the following macros have the purpose of:
% 1) changing the equation numbers to the format: (3.4) %
% 2) Labeling the Appendix Sections as A,B,... with eq: (A.3)
% The appendix starts with the declaration: \startappendix
% which resets the new counter used here for the section numbers.
%
% Changes needed for conversion to the bare-bones Latex:
% \masection => \section
% \masubsection => \subsection
% \startappendix => \appendix
% remove the space corrections next to the \section declarations
% (their purpose was to avoid a big gap with the \subsection title)
% and possibly redo the "Acknowledgment" and "Reference" lines. %
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%$$---Modified Section titles %%%%%%%%%
% invoke by:
% \masect{Introduction} %\vspace{-.6cm} --before first masubsect.
% \masubsect{Introduction}
% \maappendix{Title}
% be sure to place, just above the first appendix the line:
% \startappendix -- to restart the section counter
% any subsections of Appendices - do by hand (or modify the code)
\newcounter{masectionnumber}
\setcounter{masectionnumber}{0}
\newcommand{\masect}[1]{\setcounter{equation}{0}
\refstepcounter{masectionnumber} \vspace{1truecm plus 1cm} \noindent
{\large\bf \arabic{masectionnumber}. #1}\par \vspace{.2cm}
\addcontentsline{toc}{section}{\arabic{masectionnumber}. #1}
}
\renewcommand{\theequation}
{\mbox{\arabic{masectionnumber}.\arabic{equation}}}
\newcounter{masubsectionnumber}[masectionnumber]
\setcounter{masubsectionnumber}{0}
\newcommand{\masubsect}[1]{
\refstepcounter{masubsectionnumber} \vspace{.5cm} \noindent
{\large\em \arabic{masectionnumber}.\alph{masubsectionnumber} #1}
\par\vspace*{.2truecm}
\addcontentsline{toc}{subsection}
{\arabic{masectionnumber}.\alph{masubsectionnumber}\hspace{.1cm} #1}
}
%%%%%%%%%%%% appendix sections:
\newcommand{\startappendix}{ \setcounter{masectionnumber}{0} }
%%resetsection counter
\newcommand{\maappendix}[1]{
\setcounter{equation}{0}
\refstepcounter{masectionnumber} \vspace{1truecm plus 1cm} \noindent
{\large\bf \Alph{masectionnumber}. #1}\par \vspace{.2cm}
%% Equations in the form (A.1)
\renewcommand{\theequation}
{\mbox{\Alph{masectionnumber}.\arabic{equation}}}
\addcontentsline{toc}{section}{\Alph{masectionnumber}. #1}
}
% any subsections of Appendices - do by hand (or modify the code)
%%%%%%%%%%%%%%%%% Numbering %%%%%%%%%%
% Separate numbering of main theorems.
\newtheorem{Thm}{Theorem}
% Consecutive numbering of lemmas, props., coros., in the form section.lemma
\newtheorem{lem}{Lemma}[masectionnumber]
\newtheorem{thm}[lem]{Theorem}
\newtheorem{prop}[lem]{Proposition}
\newtheorem{cor}[lem]{Corollary}
\newtheorem{df}[lem]{Definition}
\newtheorem{rem}[lem]{Remark}
\newtheorem{claim}[lem]{Claim}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end inserted macros %%%%%%%%%%%
%%%%%%%%% That's it --- here we go:
\begin{document}
\title{\vspace*{-.35in}
On the Stability of the Quenched State
in Mean Field Spin Glass Models }
%%%%%%%%%%%%%%%%%%%%%%
\author{M. Aizenman ${}^{(a)}$
\qquad and \qquad P. Contucci ${}^{(b)}$\\ \hskip 1cm
\vspace*{-0.05truein} \\
\normalsize \it ${}^{(a)}$ School of Mathematics,
Institute for Advanced Study, Princeton NJ 08540.
\thanks{Permanent address: Departments of Physics and
Mathematics, Jadwin Hall,
Princeton University, P. O. Box 708, Princeton, NJ 08544.} \\
\normalsize \it ${}^{(b)}$ Department of Physics,
Princeton University, Princeton NJ 08544. }
\maketitle
\thispagestyle{empty} %removes # on p.1
\begin{abstract}
While the Gibbs states of spin glass models have been noted
to have an erratic dependence on temperature, one may expect
the mean over the disorder to produce a continuously varying
``quenched state''.
The assumption of such continuity in temperature
implies that in the infinite volume limit the state
is stable under a class of deformations of the Gibbs measure.
The condition is satisfied by the Parisi Ansatz, along with
an even broader stationarity property. The stability conditions
have equivalent expressions as marginal additivity of the
quenched free energy. Implications of the continuity assumption
include constraints on the
overlap distribution, which are expressed as the vanishing of the
expectation value for an infinite collection of multi-overlap
polynomials. The polynomials can be computed with the aid of a
{\it real}-replica calculation in which the number of replicas
is taken to zero.
\end{abstract}
\vskip .25truecm
%\noindent {\bf PACS numbers:} \\
\noindent {\bf Key words:} Mean field, spin glass, quenched state,
overlap distribution, replicas.
\vfill
%\newpage
% \vskip .25truecm
% \begin{minipage}[t]{\textwidth}
% \tableofcontents
% \end{minipage}
% \vskip .5truecm
\newpage
\vspace{-1.2cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Name: 1introduction.tex
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\masect{Introduction} \vspace{-.6cm}
\vskip .5truecm
We consider here the quenched state of the Sherrington-Kirkpatrick (SK)
spin glass model, and discuss some stationarity properties which
seem to emerge in the infinite volume limit.
The SK spin glass model \cite{SK} has spin variables $\sigma_i=\pm 1$,
$i = 1,\ldots, N$, interacting via the Hamiltonian
\be
H_N(\sigma,J) \ = \
- {1\over \sqrt{N}}
\sum_{1\le i$, and in case
of possible confusion by $< - >_J $, the expectation value over
the spins averaged with respect to the Gibbs state. An average over
the couplings is denoted by $Av( - )$. The combined
{\em quenched average} is a double average,
denoted below by $E( - )$, over the spins and the disorder
(whose probability distribution is not affected
[in the quenched case] by the response of the spin system
to the random Hamiltonian).
Quantities of interest include:
\bea
E\left( q_{\sigma,\sigma'}^2 \right) \ & = &\
\frac{1}{N^2} \sum_{i,j} Av\left( \ <\sigma_i \sigma_j>^2_J \ \right)
\ = \ Av\left( \ <\sigma_1 \sigma_2>^2_J \ \right) + O(\frac{1}{N})
\; , \nonumber \\
\mbox{and} \; \; \;
& & \sqrt{N} Av\left(<\sigma_1 \sigma_2> \ J_{2,3} \ <\sigma_3 \sigma_1>
\right) \; .
\label{eq:example}
\eea
The second example is seen among other terms in
${\partial \over \partial \beta}E\left( q_{\sigma,\sigma'}^2 \right)$.
Through integration by parts over the normalized gaussian variables $J$.
\be
Av\left( J f(J) \right) \ = \
Av\left( \frac{\partial}{\partial J} f(J) \right) \, ,
\label{eq:parts}
\ee
such expressions can be reduced to
averages of products of the overlaps $q_{\sigma, \sigma'}$
over a number of replicas.
In order not to overburden the notation we resist introducing
the average over a direct sum of different number of replicas.
Instead, we denote somewhat flexibly an average over their suitable
collections (indexed by $j$) by the symbol
\be
\ll - \gg=\otimes_{j}< - >^{(j)},
\label{infpr}
\ee
and we let
\be
E( - )=Av\left( \ll - \gg \right) \ .
\label{misura}
\ee
Finally, as will be seen next, we shall
add to the objects under consideration
certain Gaussian random fields, indexed by the spin configuration.
We extend the symbols $Av( - )$ and $E( - )$ to include averages
over all such fields.
\masubsect{Invariance of the Parisi solution}
Let $h(\sigma)$ and $K(\sigma)$ be two Gaussian random fields,
defined over the spin configuration, which are mutually
independent, and independent of the random couplings (i.e.,
independent of the Hamiltonian), with the covariances
\bea
Av\left( h(\sigma) h(\sigma') \right) \ & = & \ q_{\sigma, \sigma'}
\label{eq:h} \\
Av\left( K(\sigma) K(\sigma') \right) \ & = & \ q^2_{\sigma,
\sigma'} \; .
\label{eq:K}
\eea
Such quantities are motivated below: $h(\sigma)$
as the cavity field associated with an increase in $N$,
and $K(\sigma)$ as representing the change in the action
corresponding to an increase in the temperature.
The Parisi solution has the property that quenched averages
are not affected by the addition to the action of terms of
the form $F(K(\sigma), h(\sigma) )$, where $F(\cdot, \cdot)$
is any smooth bounded function.
To express the above stated property, let us denote the deformed
states as
\bea
< - >_{F(K,h)}\ & := & \ {< - \exp{\{F(K,h)\}}>\over
<\exp{\{F(K,h)\})}>} \; , \nonumber \\
\mbox{ } \nonumber \\
\ll - \gg_{F(K,h)} \ & = \ & \otimes_{i}< - >^{(i)}_{F(K,h)}
\; ,
\label{dinfpr}
\eea
and
\be
E_{F(K,h)}( - ) \ = \ Av(\ll - \gg_{F(K,h)}) \; .
\label{dmisura}
\ee
\begin{claim}
For the Parisi solution, in the infinite volume limit at any temperature:
\be
E_{F(K,h)}( - ) \ = E( - ) ,
\label{eq:parisi}
\ee
when the expectation value functionals are restricted to quantities which do
not involve the deformation fields $K$ and $h$.
\end{claim}
We shall not verify this statement here (The reader is invited
to do so from the solution, which is discussed in~\cite{MPV}
and references therein) but instead discuss the origin and
consequences of a somewhat restricted invariance of this kind.
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Name: 3continuity.tex
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\masect{Continuity in the temperature and stability under deformation}
The broad stability of the quenched state expressed by \eq{eq:parisi}
has not yet been rigorously derived for the SK model. We shall
now find that a somewhat restricted version of this condition follows
from a natural continuity assumption.
There is a significant difference between the spin--glass and
the ferromagnetic spin models in the effect of a change in
the temperature on the equilibrium state.
Reduction in the temperature amounts to increased coercion towards
the low energy states of the system.
If the ground state is unique, it is natural to expect the
equilibrium state to vary continuously at low temperatures.
When there are only few ground states, one may expect
some discontinuities (as in the Pirogov-Sinai theory
\cite{PS}). However, when there is a high multiplicity
of competing low energy states the result may be quite different.
Indeed it is reported
that for a given realization
of the random Hamiltonian, the equilibrium state has a very erratic
dependence on the temperature. Nevertheless it may seem
reasonable to expect
that with the average over the disorder, the quenched state
might vary continuously with $\beta$.
To illuminate the consequences of the continuity assumption, let us
note that due to the addition law for independent Gaussian variables,
the Gibbs factor determining the equilibrium state at the
inverse--temperature $\beta + \Delta \beta$ can be presented as
a sum of two independent terms:
\be
(\beta + \Delta\beta) H(\sigma,J) \
\mathrel{\mathop{=}\limits^{\cal D}} \
\beta H(\sigma,J) + \delta(\beta) H(\sigma,\tilde{J}) \; ,
\label{eq:deltaH}
\ee
where $A \mathrel{\mathop{=}\limits^{\cal D}} B$ means that $A$ and
$B$ have equal probability distributions,
$\{ J, \tilde{J}\}$ are two independent sets of couplings, and
\be
\delta(\beta)=\sqrt{2{\beta\Delta\beta}+({\Delta\beta})^2}
\ee
With the action cast in the form \eq{eq:deltaH},
the modified state is seen to incorporate the
effects of a strong term (of the order of the volume)
pulling in some randomly chosen
directions, when the main term itself has many competing states.
The assumption of the continuity of the quenched state appears now
as less obvious, and it should therefore carry some
notable consequences.
A related observation can be made by considering the effects of
deformation of the state through the addition of a Gaussian field of
the type $K(\sigma)$ (\eq{eq:K}). An easy computation
based on the fact that
\be
H(\sigma,J) \mathrel{\mathop{=}\limits^{\cal D}} \sqrt{N} K(\sigma)
\label{eq:HK}
\ee
shows that the effect is equivalent in a change of the order $O(\frac{1}{N})$
in the temperature:
\be
E_{\lambda K}^{(N, \beta)}( - )=E^{(N, \beta+{\lambda^2\over 2N\beta})}( - )
\; \; ,
\label{propri}
\ee
where the superscripts refer to the size
and the inverse--temperature, the subscript indicates a deformation
of the state in the sense of \eq{dinfpr} and, as for the \eq{eq:parisi}
the equality is understood when the two measures are restricted to the
quantities independent from the deformation variable $K$.
In the limit $N \to \infty$, the change in the temperature on the
right side vanishes.
This immediately leads to the following observation.
\begin{prop}
If, over a certain temperature range,
the states $E^{(N, \beta)}(-)$ are uniformly continuous in $beta$,
as $N\to \infty$, and the
infinite--volume limit exist for the quenched state
then the limiting
measures $E^{(\beta)}(-)$ are stable under linear deformations
in $\lambda K$, in the sense:
\be
E^{(\beta)}\left( - \right) \ =\ E^{(\beta)}_{\lambda K}\left( - \right).
\label{eq:stability}
\ee
restricted to the algebra of observables generated by all
the random variables but the deformation field $K$
\label{thm:stability}
\end{prop}
Let us note that the assumptions made above imply also
another stationarity principle: the quenched state would
be invariant under the deformation induced by
$\ln 2\cosh\beta h$. To see that, compare the state of $N+1$
particles with that of $N$. The trace over the ``last'' spin
yields for
expectation values of functions of the ``first'' $N$ spins
\be
E^{(N+1, \beta)}\left( - \right) \ =\
E^{(N, \tilde{\beta})}_{\ln 2\cosh\tilde{\beta}h}\left( - \right) \; ,
\label{propri2}
\ee
where $\tilde{\beta}=\sqrt{N\over N+1}\beta$ and $h(\sigma)$ is a
Gaussian field with covariance $q_{\sigma,\sigma '}$.
Under the stated assumptions, in the
thermodynamical limit the previous relation becomes:
\be
E\left( - \right) \ =
\ E_{\ln 2\cosh\beta h}\left( - \right) \; .
\label{propri3}
\ee
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Name: 4logarithmic.tex
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\masect{A logarithmic relation expressing the stability property}
The stability condition which follows from the above discussed
continuity assumption is indeed found among the properties of the
Parisi solution,
along with the more sweeping stability property expressed in
~\eq{eq:parisi}. In this section we cast the broader
invariance property in the form of a logarithmic relation,
which expresses an additivity property for the marginal
increments in the quenched free energy.
\begin{df} We say that a random system, in the quenched state
$Av\left( \ll - \gg \right)$, has marginally-additive free energy
if for any finite collection
of independent Gaussian fields $K^{(1)}, K^{(2)},\ldots, K^{(l)}$
with the covariance \eq{eq:K}, and any smooth polynomially
bounded functions $F_1, F_2,\ldots, F_l$
\be
Av \ln <\exp(\sum_{i=1}^{l} F_i(K^{(i)}))> =
\sum_{i=1}^{l} Av \ln <\exp F_i(K^{(i)})>
\label{eq:log}
\ee
\end{df}
Our main observation is that the above marginal additivity
of the quenched free energy
is equivalent to the {\em stability} of the quenched state
expressed by
\be
E\left( - \right) \ =E_{F(K)}\left( - \right) \ ,
\label{9}
\ee
where $F$ is an arbitrary smooth bounded function,
and the equation means
equality of the expectation values of quantities involving
functions of the spins, random couplings, and auxiliary Gaussian
fields -- other than the deforming field $K$.
Let us note that
the expectation values of quantities involving any of the above,
can be evaluated by first integrating over the extraneous
Gaussian variables ($K$). Using Wick's formula, this
integration produces expressions involving the overlaps among
arbitrary number of replicas, as in:
\bea
E\left( K_1\, K_2\, K_2'\, K_3'\right)\
&=& \ Av\left( \right)
\nonumber \\
& = & \
E(q^2_{1,2}q^2_{2,3}) \; ,
\label{eq:dict}
\eea
where we defined $K_i=K(\sigma^{(i)})$
Conversely, the averages of polynomials in replica overlaps,
as $q^2_{1,2}q^2_{2,3}$ in the above expression, can be easily
expressed through the expectation values of suitable products of
independent copies of the $K$ field. Based on this observation
one can see that the measure $E(-)$ is completely determined
by the distribution of the overlaps.
Clearly the stability implies the marginal additivity property
(of the free energy).
To prove the converse, we need to show that
\be
Av \left[ _{F(K')} \right]^n \ = \
Av \left[\right]^n \; ,
\label{eq:power}
\ee
for any integer $n$ and polynomial function $G$.
(The full statement takes a bit more general form
-- involving
products with different functions $G$ for the different copies
of the spin system, however by the polarization argument there
is no loss of generality in taking the same function $G$
for all the $n$ replicas.)
Let us note also that, by an elementary approximation argument,
it suffices to prove \eq{eq:power} for bounded functions $G$.
The logarithmic property (\ref{eq:log}) implies that for all
$\varepsilon$ the following function vanishes
\be
\varphi(\varepsilon) \equiv=
Av \ln {{<\exp(\varepsilon G + F)>}
\over{<\exp(\varepsilon G)><\exp(F)>}} \ = \ 0 \; .
\label{10}
\ee
For bounded $G$ the function $\varphi(\varepsilon)$ is analytical
in a strip containing the real axis, and the logarithmic
property \eq{eq:log} is equivalent to the vanishing of
all the derivatives of $\varphi(\varepsilon)$ at $\varepsilon=0$.
By an inductive argument, these conditions imply \eq{eq:power}:
first we observe that
\be
\varphi'(\varepsilon)|_{0}= Av(_F)-Av()=0,
\label{indu1}
\ee
which is the stability for the first power.
The second derivative gives
\be
\varphi''(\varepsilon)|_{0}= Av(^2_F)
-Av(_F)-Av(^2)+Av()=0 \; .
\label{indu2}
\ee
Thus
\bea
Av(^2_F)- Av(^2)\ & = & \ Av(_F) - Av()
\nonumber \\
\ & =& \ 0 \; ,
\label{indu3}
\eea
where the last equality is by the first order equation, \eq{indu1}, applied
to the smooth function $G^2$.
Continuing in this fashion one may see that if stability is fulfilled up
to power $n$ it is fulfilled for power $n+1$.
\begin{rem}
The truncated expectations (cumulants) of order $p$ are generally defined by
\be
<-;p>^T=
{\partial^p \over \partial\lambda^p}\ln
<\exp (\lambda -)>|_{\lambda=0} \; .
\label{61}
\ee
In these terms the logarithmic relation is equivalent to:
\be
Av <\sum_{i=1}^{l}F_i(K^{(i)});p>^{T} \ =\
Av \sum_{i=1}^{l} ^{T}
\label{eq:truncated}
\ee
for every integer $p$. (The implication \eq{eq:log}
$\Rightarrow$ \eq{eq:truncated} is obvious.
In the other direction the proof can be based on the analyticity
argument indicated above.)
It might be noted that an equation like \eq{eq:truncated} cannot
possibly hold without the average {\em Av}, unless the Gibbs
state is
typically supported on a narrow collection of configurations
over which the overlap function
takes only the value $q_{\sigma,\sigma'}=1$.
\end{rem}
Equations (\ref{eq:log}) and (\ref{eq:truncated}) have natural
counterparts for the more limited stability of the quenched state,
expressed by \eq{eq:stability}. In that case
$F_i(K^{(i)})$ need by replaced by $\lambda_i K^{(i)}$, for
$i=2,3,\ldots$ (thought $F_1$ may still be left arbitrary.)
It is an interesting open question whether there are states
stable in the limited sense which do not fulfill the stronger
stability condition.
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Name: 5overlap.tex
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\masect{Overlap polynomials with zero average}
We now turn to some of the implications of the stability condition
\eq{eq:stability} which was shown to follow from the continuity assumption.
Since the free energy and its derivatives are determined through the
distribution of the overlaps, it is natural to ask what consequences
does the stability property have in those terms.
As we shall see next, the implications include a family of relations
expressed as the vanishing of the expectation value of
an infinite collection of overlap polynomials.
>From a combinatorial point of view, expressions with vanishing
expectation are constructed by applying a certain
operation to graphs representing overlap polynomials.
We shall use the notation encountered already in \eq{eq:dict},
where $q_{1,2}$
indicates the overlap between two spin configurations sampled from two
different copies of the system, one in replica $1$ and the other in the
replica $2$, subject to the same random interaction.
Products of such
terms can be represented by labeled graphs, introduced below.
The expectation value does not depend on the particular labeling
of the different replicas, for instance
\be
E(q^2_{1,2}q^4_{2,3}q^2_{1,4})=E(q^2_{1,2}q^2_{2,3}q^4_{3,4}) \; .
\label{12}
\ee
Let now illustrate some consequences of the stability
condition, starting from the easy example
\be
Av(^2_{\lambda K'})=Av(^2).
\label{13}
\ee
The first derivative with respect to lambda gives
\be
2Av(_{\lambda K'}(_{\lambda K'}-_{\lambda K'}_{\lambda K'}))=0
\label{a}
\ee
which vanishes, for the trivial reason of parity, at $\lambda=0$.
On the other hand, the second derivative yields
\begin{eqnarray}
2Av(^2_{\lambda K'}-4_{\lambda K'}_{\lambda K'}_{\lambda K'}
+3^2_{\lambda K'}^2_{\lambda K'}\nonumber \\
+_{\lambda K'}_{\lambda K'}
-^2_{\lambda K'}_{\lambda K'})=0.
\label{14}
\end{eqnarray}
At $\lambda = 0$ the last two terms cancel and the expression
reduces to:
\be
2E(q_{1,2}^4)-8E(q^2_{1,2}q^2_{2,3}) +6E(q^2_{1,2}q^2_{3,4})=0 \; .
\label{15}
\ee
As was mentioned already, the stability property is satisfied by the Parisi
solution, and hence this relation, as well as those of higher order
derived below, are satisfied there.
The particular case of, \eq{15} was
recently derived (for almost every $beta$) without any assumptions
in ref.~(\cite{G}).
One may note, that \eq{15} is also the lowest non-trivial
identity of those listed in \eq{eq:truncated} (corresponding to
$p=4$, $F(K)\equiv K$).
The previous relation tells us that the polynomial
\be
q_{1,2}^4-4q^2_{1,2}q^2_{2,3} +3q^2_{1,2}q^2_{3,4}
\label{poly}
\ee
has zero average. Let us present now a systematic approach for the
derivation of other such conditions.
One may use a graphical representation in which a monomial of the
form $q^2_{1,2}q_{2,3}$ is identified with a graph whose vertices
are the replica indices $\{1, 2, 3\}$ and the edges correspond to the
overlaps, $q_{i,j}$.
Such a graph will be indicated by the symbol
$(1,2)^2(2,3)$. Furthermore, we shall consider also products
involving an additional Gaussian field ($K$). The graphical
representation of that factor is a half-edge, represented
by a singleton. I.e., $(1,2)(2)$ and $(1,2)(3)$ correspond to
$q_{1,2}K_{2}$ and $q_{1,2}K_{3}$.
We shall use a product ``$\cdot$'' which acts in the space of graphs
as {\em composition} combined with {\em contraction}, where possible, of
the two unpaired legs. The notion may be clarified by the following
examples:
\bea
(1,2)\cdot(1,2)(2) & =& (1,2)^2(2) \nonumber \\
(1,2)(3)\cdot(4) & = & (1,2)(3,4) \\
\eea
Terms of the form $(1,1)$ can be omitted, since in our case
$q_{1,1}=1$.
The above product turns out to be commutative but not
associative. The {\it order} of a graph is defined as
$2\times\mbox{\em number of edges}$, with half-edges counting as $1/2$.
Let $W_k$ denote the space of formal linear combination of graphs
of order $k$. For $G\in W_k$ we denote by $O_G$ the corresponding
element of the overlap algebra.
We define $\delta: W_k \rightarrow W_{k+1}$ as the linear operator
which acts on single graphs by
\be
\delta G=\sum_{v\in V(G)}\delta_v G,
\label{der}
\ee
where $V(G)$ is the set of vertices of $G$, and
\be
\delta_v G= G\cdot (v) -G\cdot (\tilde{v}),
\label{deri}
\ee
where $\tilde{v}$ is a new vertex not belonging to $G$.
E.g., $\delta_1 (1,2)= (1,2)(1)-(1,2)(3)$,
$\delta_1 (1,2)(1)=(1,2)(1)\cdot(1)-(1,2)(1)\cdot(3)=(1,2)-(1,2)(1,3)$.
Following is the pertinent observation.
\begin{prop}
For any measure $E(-)$, of the structure seen in \eq{misura},
\be
{\partial^2\over \partial \lambda^2} E_{\lambda K}(Q_G)|_{\lambda=0}=
E(Q_{\delta^2 G}) \; ,
\label{prop:16}
\ee
for all the elements $Q_G$ of the overlap algebra.
\end{prop}
The proof is given below. A direct consequence is:
\begin{prop}
If the measure $E(-)$ is invariant under linear deformations, in the
sense of \eq{eq:stability},
then for all the elements $O_G$ of the overlap algebra:
\be
E(O_{\delta^2 G})=0 \; .
\label{160}
\ee
\end{prop}
The proof of Proposition~\ref{prop:16} is straightforward.
The operation $\delta$ is the graphical counterpart of the usual derivative
with respect to the parameter $\lambda$ in the Boltzmann weight
(where it appears in $\lambda K$).
Such a derivative produces a {\it truncated correlation} expressed in
the rule (\ref{deri}). The (\ref{der}) is nothing
but the Leibnitz rule for derivative of products. The first
differentiation produces a sum of monomials, each containing an unpaired
centered Gaussian variable ($K$) of zero mean.
The second derivative produces another unpaired variable,
which is contracted
with the previous one via the Wick rule. (This contraction motivates the
product introduced above.) The fact that the second
derivative has a zero average is a nontrivial consequence of the stability.
Following is a related statement, obtained by combining \eq{propri} with
\eq{eq:stability}.
\begin{prop} In the SK model, at finite values of $N$,
\be
{\partial\over \beta\partial\beta}E_N(Q_G)=N
{\partial^2\over \partial \lambda^2} E_{N,\lambda K}(Q_G)|_{\lambda=0}=
N E_N(Q_{\delta^2 G}),
\label{17}
\ee
for every element of the overlap algebra. In particular, if the quenched
state is locally differentiable in $\beta$ uniformly in $N$, then
the expectation values of
quantities of the form $Q_{\delta^2 G}$ vanish at the rate $O(1/N)$ or faster.
\end{prop}
The last statement supplements the previous assertion with a somewhat
stronger conclusion (suggesting a numerical test), which is
derived under a stronger assumption.
Identical result holds for other mean field spin glass
models, with the $p$-spin interaction Hamiltonian (\cite{D})
\be
H(\sigma,J)=
-\sum_{1\le i_1<\cdots< i_p\le N}J_{i_1,\cdots,i_p}\sigma_{i_1}\cdots
\sigma_{i_p},
\label{genep}
\ee
where $J_{i_1,\cdots,i_p}$ are Gaussian variables, rescaled
so that the Hamiltonian covariance is
\be
Av H(\sigma,J)H(\sigma',J)=N q^p_{\sigma,\sigma'}.
\label{pcova}
\ee
(Under the above calling, $H$ reaches values of order $N$.
The corresponding choice for the field $K$ is Gaussian with the
covariance
$Av(K(\sigma)K(\sigma '))=q^p_{\sigma, \sigma '}$.
The definition of $\delta$ is unchanged.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%r
%
% Name: 6replica.tex
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\masect{Computation with real replicas}
In this section we give several characterizations of the
overlap polynomials of the form $Q_{\delta^2 G}$, for which we saw that
stability condition implies zero mean.
The main result is a formula
which permits to compute the polynomials from a quadratic expression
in the number $r$ of {\it real}-replicas, evaluated at $r=0$.
To state it, let $M_r=\sum_{i\ne j=1}^{r}q^2_{i,j}$, for all integers
$r\ge 1$. Let $E(-)$ be an expectation value functional,
on the algebra of overlaps, which depends only on the
graph structure of the overlap monomials (i.e., is independent of
the choice of labels). Then the quantity $E(Q_G M_r)$ is quadratic
in $r$. In the following proposition, we refer to the {\em
polynomial extension} of this function to all real $r$.
\begin{prop} For any expectation value functional $E(-)$, as above,
\be
E(Q_{\delta^2 G})=E(Q_G M_r)|_{r=0}.
\label{eq:replica}
\ee
where the quantity $E(Q_G M_r)$ is first computed
for $r$ large enough so that all the indices appearing in
$Q_G$ do appear also in $M_r$ and $r>|G|+1$.
\label{prop:replica}
\end{prop}
To illustrate the statement, let us take: $G=q^2_{1,2}$.
In this case, the left side of \eq{eq:replica}
is given by \eq{15} and the right side is:
\be
E(q^2_{1,2}M_r)=2E(q^4_{1,2})+4(r-2)E(q^2_{1,2}q^2_{2,3})
+(r-2)(r-3)E(q^2_{1,2}q^2_{3,4}) \; .
\label{exem}
\ee
The two coincide at $r=0$ (defined by polynomial extension).
The proof proceeds through the explicit computation of
the left and right sides of \eq{eq:replica}.
\begin{lem}
If the number of vertices in $G$ is $l$ then
\be
\delta^2 G = \sum_{v\ne v'}G\cdot (v,v')-2l\sum_{v}G\cdot (v,\tilde{v})+
l(l+1)G\cdot (\tilde{v},\tilde{v}'),
\label{espgr}
\ee
where $v$ and $v'$ are summed over the set of vertices of
$G$ and $\tilde{v}$ and $\tilde{v}'$ denote a pair of added vertices.
\end{lem}
This is a rather explicit expression for the polynomials corresponding
to a given graph $G$. Two examples are:
\bea
\delta^2 (1,2)(3,4) & = &
4(1,2)^{2}(3,4)\ +\ 8(1,2)(2,3)(3,4) -
\nonumber \\
& & -32(1,2)(2,3)(4,5) \ + \ 20(1,2)(3,4)(5,6) \; ,
\label{twopi}
\eea
and
\bea
\delta^2 (1,2)(2,3) & = &
4(1,2)^{2}(2,3)+2(1,2)(2,3)(3,1)\ -\ 12(1,2)(2,3)(3,4) +
\nonumber \\
& & + 12(1,2)(2,3)(4,5)\ -\ 6(1,2)(2,3)(2,4) \; .
\label{thevi}
\eea
\noindent{\bf Remark} In the above example $\delta^2 G$
is a polynomial expression with integer coefficients
whose sum is zero. That property is shared by $\delta^2 G$
of arbitrary monomials $G$.
To prove the lemma we note that by the definition of $\delta$
\be
\delta G= \sum_{v}G\cdot (v)- lG\cdot (\tilde{v}) \; .
\label{lem11}
\ee
Applying this rule twice
\bea
\delta^{2}G\ & =\ & \sum_{v,v'}G\cdot (v,v')-l\sum_{v}G\cdot (v,\tilde{v})
-l\sum_{v'}G\cdot (v',\tilde{v})-l\sum_{v'}G\cdot (\tilde{v},\tilde{v})\\
\nonumber
& & + l(l+1)G\cdot (\tilde{v},\tilde{v}'),
\label{lem12}
\eea
which coincides with \eq{espgr} since $(v,v)=1$ for every $v$.
\begin{lem}
If all the replica indices appearing in $Q_G$ are contained in the entries
of the matrix $M_r$ and $r>l+1$ then
\bea
E(Q_G M_r)\ =\ \sum_{
\begin{array}{c}
{\footnotesize v, v' \in V(G) } \\
{\footnotesize v\ne v' }
\end{array} }
E(Q_{G\cdot (v,v')})
\ + \ 2(r-l)\sum_{v\in V(G)} E(Q_{G\cdot (v,\tilde{v})}) \ \nonumber
\\
+\ (r-l)(r-l-1)E(Q_{G\cdot (\tilde{v},\tilde{v}')}).
\label{rrep}
\eea
\end{lem}
This formula is an elementary consequence of the fact that the measure
$E(-)$ depends only on the isomorphism type of the graph associated to a
given overlap monomial. The first sum on the right side of \eq{rrep}
corresponds to those overlap terms in
$M_r$ which involve only the replicas appearing in $G$, the other two
sums are split according to whether the number of vertices not in $G$
is $1$ or $2$.
The two previous lemmas prove Proposition~\ref{prop:replica}.
\hfill\blackbox
%%%%%%
\masect{Comments} \vspace{-.6cm}
\vskip .5truecm
We have seen that elementary continuity assumptions
on the quenched state, imply a stability property for the
infinite volume limit of the SK spin glass models.
A particular implication, translated
in terms of the overlap distribution,
is that in mean-field models the joint
probability distribution of the overlap meets an infinite family
of conditions, expressed as the vanishing of the expectation value for
a family of multi-overlap polynomials.
We also saw a finite volume condition expressed as a decay rate
for the expectation values of suitable quantities.
These observations are consistent with the Parisi theory.
However, the family of identities
discussed here does not yet permit the reconstruction of the joint
probability distribution from that of a single overlap, as is the
case under the Parisi Ansatz.
I. Kondor and M. Mezard pointed out that within the replica-symmetry-
breaking approach the vanishing of the expectation
values of the polynomials discussed here ($\delta^2 G$) requires
only the so called ``replica--equivalence'' assumption, which says that
in the matrix $Q$ (defined in \cite{MPV}) each row is a
permutation of any other. However, {\it replica--equivalence}
does not appear to yield a broader class of states than the
Parisi Ansatz (M. Mezard - private communication).
An interesting question is whether the stability property is the
stationarity condition for some variational principle.
This is related to the main question which emerges
at this point, which is whether stability
implies the GREM state structure \cite{R}.
We study a restricted version of this question
in a separate paper \cite{AC2}.
\noindent {\large \bf Acknowledgments\/}
This work was supported by the NSF Grant PHY-9512729 and,
at the Institute for Advanced Study, by a grant from
the NEC Research Institute.
\vspace{-.6cm}
\addcontentsline{toc}{section}{References}
\begin{thebibliography}{1}
\bibitem{SK}
D.Sherrington and S.Kirkpatrick, ``Solvable model of a spin-glass,'' {\em Phys.
Rev. Lett.}, {\bf 35}, 1792--1796 (1975).
\bibitem{MPV}
M.M\'ezard, G.Parisi, and M.A.Virasoro, {\em Spin Glass Theory and Beyond}.
\newblock World Scientific, 1987.
\bibitem{G}
F.Guerra, ``About the overlap distribution in a mean field spin glass model,''
{\em Int. J. Phys. B}, {\bf 10}, 1675--1684 (1997).
\bibitem{R}
D.Ruelle, ``A mathematical reformulation of derrida's {REM} and {GREM},'' {\em
Commun. Math. Phys}, {\bf 108}, 225--239 (1987).
\bibitem{AC2}
M.Aizenman and P.Contucci, ``The Indy-500 model,'' {\em in preparation}.
\bibitem{PS}
S.Pirogov and J.Sinai, ``Phase diagrams of classical lattice systems,'' {\em
Teoret. Mat. Fiz. (Russian)}, {\bf 25}, 358--369 (1975).
\nopagebreak
\bibitem{D}
B.Derrida, ``Random-energy model: Limit of a family of disordered models,''
{\em Phys. Rev. Lett.}, {\bf 45}, 79--82 (1980).
\end{thebibliography}
\end{document}