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PACS 75.50.Lk
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Spin glass, SK model, replica symmetry breaking,
ultrametricity, variational bounds, GREM, Sherrington Kirkpatrick model, Parisi solution, Guerra bounds, rigorous results, critical review
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%% DATE: June 14, 2003
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\def\sig{ \sigma}
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\begin{document}
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\title{An Extended Variational Principle for the
SK Spin-Glass Model}
% \draft command makes pacs numbers print
% repeat the \author\address pair as needed
\author{Michael Aizenman, Robert Sims, and Shannon L. Starr}
%%%
%%% Changed \address to \affiliation
%%%
\affiliation{
Departments of Physics and Mathematics, Jadwin Hall, Princeton
University, Princeton, NJ 08544}
\date{\today}
\begin{abstract}
The recent proof by F. Guerra that the Parisi ansatz provides a
lower bound on the free energy of the SK spin-glass model
could have been taken as offering some support to the
validity of the purported solution.
In this work we present a broader variational principle, in which
the lower bound, as well as the actual value, are expressed through
an optimization procedure for which ultrametic/hierarchal structures
form only a subset of the variational class.
The validity of Parisi's ansatz for the SK model is still in question.
The new variational principle may be of help in critical review of the issue.
\end{abstract}
\pacs{75.50.Lk}
%%%%%%%
\maketitle
%%%%%%%
\noindent{\em Introduction \qquad }
The statistical mechanics of
spin-glass models is characterized by the existence of a diverse
collection of competing states, very slow relaxation of the quenched
dynamics, and a rather involved picture of the equilibrium state.
A great deal of insight on the subject has been produced through
the study of the Sherrington Kirkpatrick (SK) model~\cite{SK}. After some
initial attempts, a solution was proposed by G. Parisi which has the
requisite stability and many other attractive features~\cite{P}. Its
development has yielded a plethora of applications of the method,
in which a key structural assumption is a particular form
of the replica symmetry-breaking (i.e., the assumption of
``ultrametricity'', or the hierarchal structure, of the overlaps among
the observed spin configurations)~\cite{MPV}.
Yet to this day it was not established that this very
appealing proposal does indeed provide the equilibrium structure of
the SK model. A recent breakthrough is the proof by
F.~Guerra~\cite{Gue} that the free energy provided by Parisi's
purported solution is a rigorous lower bound for the SK free energy.
More completely, the result of Guerra is that for any value of the
order parameter, which within the assumed ansatz is a {\em function},
the Parisi functional provides a rigorous lower bound.
Thus, this
relation is also valid for the maximizer which yields the Parisi
solution.
In this work we present a variational principle for the free energy of
the SK model which makes no use of a Parisi-type order parameter, and
which yields the result of Guerra as a particular implication. More
explicitly, the new principle allows more varied bounds on the free
energy, for which there is no need to assume a hierarchal organization
of the Gibbs state (e.g., as expressed in the assumed ultrametricity
of the overlaps~\cite{MPV}). Guerra's results follow when the
variational principle is tested against the Derrida-Ruelle hierarchal
probability cascade models (GREM)~\cite{Ru}.
This leads us to a question which is not new: is the ultrametricity
an inherent structue of the SK mean-field model, or is it only a
simplifiying assumption. The new variational principle may provide a
tool for challenging tests of this issue.
\noindent{\em The model \qquad} The SK model concerns Ising-type spins,
$\sig = (\sigma_1,\dots,\sigma_N)$,
with an a-priori equi-distribution over the values
$\{ \pm 1\}$, and the random Hamiltonian
\begin{equation}
H_{N}(\sig)\ = \ \frac{-1}{\sqrt{N} }
\sum_{1\le i < j \le N} J_{ij} \, \sigma_{i}\sigma_{j}
\ - \ h \sum_{i=1}^N \sigma_{i}
\end{equation}
where $\{J_{ij}\}$ are independent normal Gaussian variables.
Our analysis applies to a more general class of Hamiltonians which
includes all the even ``$p$-spin'' models~\cite{Der,GT2}. Namely:
\begin{equation}
H_N(\sig)\ =
\ - K_N(\sig) \ - \ h \sum_{i=1}^N \sigma_{i}
\end{equation}
with
\begin{equation} \label{eq:pspin}
K_N(\sig)\ = \ \sqrt{ \frac{N}{2} } \,
\sum_{r=1}^\infty \frac{a_r}{ N^{r/2}} \sum_{i_1,\dots,i_r=1}^N
J_{i_1\dots i_r} \sigma_{i_1}\cdots\sigma_{i_r}
\end{equation}
where all the $\{J_{i_1,\dots,i_r}\}$ are independent normal Gaussian
variables (for convenience, the tensor is not assumed here to be
symmetric),
and $\sum_{r=1}^{\infty} |a_r|^2 =1$.
As in \cite{GT2}, our argument requires that the function
$ f(q) \ = \ \sum_{r=1}^{\infty} |a_r|^2 \, q^r \, $
be convex on $[-1,1]$.
One may note that $K_N(\sig)$ form a family
of centered Gaussian variables with the covariance
\begin{equation}
\Ev{ K_N(\sig)\, K_N(\sig')\ \vert\
\sig\, ,\, \sig'}
\ = \ \frac{N}{2} \, f( q_{\sig,\sig'}) \, ,
\label{eq:fq}
\end{equation}
which depends on the spin-spin overlap:
$q_{\sig, \sig'} = \frac{1}{N} \sum_j
\sigma_j \sigma'_j$.
The standard SK model corresponds to $f(q)=q^2$.
The partition function, $Z$, the quenched free energy, $F$, and
what we shall call here the pressure, $P$, are defined as
\begin{gather}
Z_{N}(\beta,h) =
\sum_{\sigma_1,\ldots, \sigma_N = \pm 1} e^{-\beta\
H_{N}(\sig)}
\label{eq:Z} \\
P_{N}(\beta,h) \ = \ \frac{1}{N}
\Ev{ \log Z_{N}(\beta,h) } \ = \ - \beta\ F_{N}(\beta,h)
\end{gather}
where $\Ev{-}$ is an average over the random couplings $\{J_{ij}\}$.
The thermodynamic limit for the free energy, i.e., the existence
of $\lim_{N\to \infty}P_{N}(\beta,h) = P(\beta,h)$, was recently
established by Guerra-Toninelli~\cite{GT} through a much-awaited
argument.
\medskip
\noindent{\em The Variational Principle \quad }
Our variational expression for
$P(\beta,h)$ employs a setup which may at first appear strange, but is
natural from the cavity perspective, when one
considers the change in the total free
energy caused by the addition of $M$ spins to a much larger system of size $N$.
The expression for $Z_{N+M}/Z_{N}$ simplifies in the limit $N\to \infty$,
at fixed $M$. In the following idealized definition one may
regard the symbol $\alpha$ as representing the configuration of the
bulk. The discretness seen in the definition ($\sum \xi_{\alpha}$)
is just for the convenience of the formulation of the variational bounds,
and not an
assumption on the Gibbs state, though such an assumption may well be
true. (A more general formulation is possible,
but not much is lost by restricting attention to the
``ROSt'' defined below.)
% \begin{defn}
\noindent {\bf Definition (Random Overlap Structures):} { A {\em
random overlap structure} (ROSt) consists of a probability space
$\{\Omega, \mu(d\omega) \} $ where for each $\omega$ there is
associated a system of weights $\{ \xi_{\alpha}(\omega)\}$ and an
``overlap kernel'' $\{ \tilde{q}_{\alpha, \alpha'}(\omega) \} $ such that,
for
each $\omega \in \Omega$,
\begin{enumerate}
\item[i.] $\sum_{\alpha} \xi_{\alpha}(\omega) \ \le \ \infty $,
\item[ii.] the quadratic form corresponding to
$\{ \tilde{q}_{\alpha, \alpha'}\}$ is positive definite,
\item[iii.] $\tilde{q}_{\alpha, \alpha}=1$, for each
$\alpha$, and hence (by the Schwarz inequality) also:
$|\tilde{q}_{\alpha, \alpha'}|\le 1$ for all pairs $\{ \alpha,
\alpha'\}$.
\end{enumerate}
}
% \end{defn}
An important class of ROSt's is provided by the Derrida-Ruelle
probability cascade model which is formulated in ref.~\cite{Ru}
(called there GREM).
Without additional assumptions, one may associate to the points in
any ROSt two independent families of centered Gaussian variables
$\{\eta_{j,\alpha} \}_{j=1,2,\ldots}$ and $\{\K_{\alpha}\}$
with covariances (conditioned on the random configuration
of weights and overlaps)
\begin{gather} \label{eq:etacov}
\Ev{\, \eta_{j,\alpha} \eta_{j',\alpha'} \, |
\,
\tilde{q}_{\alpha, \alpha'}
\, }
\ = \ \frac{1}{2} \delta_{j,j'} \ f'(\tilde{q}_{\alpha,
\alpha'}) \, ,\\ \label{eq:kapcov}
\Ev{\, \K_{\alpha} \K_{\alpha'} \, |
\,
\tilde{q}_{\alpha, \alpha'}
\, }
\ = \ \tilde{q}_{\alpha, \alpha'} f'(\tilde{q}_{\alpha, \alpha'}) -
f(\tilde{q}_{\alpha, \alpha'}) \, .
\end{gather}
The existence of such processes requires positive-definiteness of the
joint covariance, but that is evident from the following explicit
construction in the case that the $\alpha$'s are
$N$-vectors, with $q_{\alpha,\alpha'}=\frac{1}{N}\sum_j
\alpha_j,\alpha_{j}'$:
\begin{equation}
\eta_{j,\alpha} \ = \ \sqrt{\frac{N}{2}} \sum_r \frac{\sqrt{r}
\, a_r}{N^{r/2}}
\sum \,
\widetilde{J}_{j,i_1,\ldots,i_{r-1}}
\alpha_{i_1}\cdots\alpha_{i_{r-1}}
\end{equation}
where the second sum is over $i_1,\ldots,i_{r-1}$ which range from
$1$ to $N$, and
\begin{equation}
\K_{\alpha} \ = \ \sum_r \frac{\sqrt{r-1}\, a_r}{N^{r/2}}
\sum_{i_1,\ldots,i_r=1}^{N} \,
\widehat{J}_{i_1,\ldots,i_r} \alpha_{i_1}\cdots\alpha_{i_r}
\end{equation}
We shall now denote by $\E(\cdot)$ the combined average,
which corresponds to integrating over three sources of randomness:
the SK random couplings $\{ J_{ij} \}$,
the random
overlap structure described by the measure $\mu(d\omega)$, and
the Gaussian variables $\{\K_{\alpha}\}$ and $\{ \eta_{j,\alpha} \}$.
Guided by the cavity picture, we associate with each
ROSt the following quantity:
\begin{multline} \label{ew:G}
G_M(\beta,h; \mu) \ = \\
= \frac{1}{M} \ \Ev{ \, \log \left(
\frac{\sum_{\alpha, \sig} \xi_{\alpha}
\, e^{\beta \sum_{j=1}^{M} (\eta_{j,\alpha} + h ) \sigma_{j} } }
{\sum_{\alpha} \xi_{\alpha} \, \, e^{\beta \sqrt{M/2}
\, \K_{\alpha} } }\right) }
\end{multline}
where $\sig = (\sigma_1,\ldots, \sigma_M) $
Our main result is:
% \begin{thm}
\noindent{\bf Theorem 1}
{\em {\em i.} For any finite $M$,
\begin{equation}
P_{M}(\beta,h) \ \le \ \inf_{(\Omega, \mu)} \
G_M(\beta,h; \mu) \ \le \ P_{U}(\beta,h) \, ,
\label{eq:AFN}
\end{equation}
where the infimum is over random overlap structures (ROSt's)
and $P_{U}(\beta,h)$ denotes the free energy $\times(-\beta) $
obtained through the Parisi ``ultrametric'' (or ``hierarchal'')
ansatz. \\
{\em ii.} The infinite volume limit of the free energy
satisfies:
\begin{equation}
\label{eq:BFN}
P(\beta,h) \ = \ \lim_{M\to \infty}
\inf_{(\Omega, \mu)} \ G_M(\beta,h; \mu) \, \, .
\end{equation}
}
% \end{thm}
% \begin{proof}
\noindent{\em Proof \quad }
These results can be seen as
consisting of two separate parts: lower and an upper bounds,
which are derived by different arguments.
{\em i.} The upper bound: the left inequality in \eq{eq:AFN},
employs an interpolation argument which is akin to that used in the
analysis of Guerra~\cite{Gue}, but which here is formulated in
broader terms without invoking the ultrametric ansatz. The second
inequality in (\ref{eq:AFN}) holds since the Parisi calculation represents the
restriction of the variation to the subset of hierarchal ROSt's.
To derive the first inequality let us introduce a family of
Hamiltonians for a mixed sytem of $M$ spins
$\sig = (\sigma_1,\dots,\sigma_M)$ and the ROSt variables $\alpha$,
with a parameter $0\le t\le 1$:
\begin{multline}
- H_{M}(\sig, \alpha; t) =
\frac{\sqrt{(1-t)M}}{\sqrt{2}}(K_{M}(\sig) + \K_\alpha)\ + \\
+\ \sqrt{t} \sum_{j=1}^{M} \eta_{j,\alpha} \sigma_{j}
\ +\ h \sum_{j=1}^M \sigma_j\, ,
\end{multline}
and let
\begin{equation}
P_M(\beta,h;t) \ = \ \frac{1}{M} \ \Ev{ \, \log \left(
\frac{\sum_{\alpha, \sig} \xi_{\alpha}
\, e^{ - \beta H_{M}(\sig, \alpha; t) } }
{\sum_{\alpha} \xi_{\alpha} \, \, e^{\beta \sqrt{M/2}
\, \K_{\alpha} } } }
\right)\, .
\end{equation}
Then
\begin{align}
P_M(\beta,h;0) &= P_M(\beta,h)\, , \label{eq:int0} \\
P_M(\beta,h;1) &= G_M(\beta,h; \mu)\, ,
\label{eq:int1}
\end{align}
and we shall show that $ \frac{d}{dt}P_M(\beta,h;t) \ \ge \ 0$.
We use the following notation for replica averages over
pairs of spin and ROSt variables. For any
$X=X(\sig,\alpha)$ and $Y=Y(\sig,\alpha;\sig',\alpha')$:
\begin{eqnarray}
\E^{(1)}_{t}(X) & = &\Ev{\ \sum_{\alpha,\sig}
\ w(\sig,\alpha;t)\ X \ } \\
\Ert{Y} \
& = & \ \Ev{\ \sum_{\alpha,\sig}
\sum_{\alpha',\sig'}\ w(\sig,\alpha;t)\
w(\sig',\alpha';t) \ Y \ } \nonumber
\end{eqnarray}
with the ``Gibbs weights''
\begin{equation}
w(\sig,\alpha;t) =
\xi_{\alpha} e^{-\beta H_{M}(\sig,\alpha;t)} \Big/
\sum_{\alpha,\sig} \xi_{\alpha}
e^{-\beta H_{M}(\sig,\alpha;t)} \, .
\end{equation}
We now have
\begin{equation}
\frac{d}{dt}P_M(\beta,h;t) \ =
- \frac{\beta}{M} \E_t^{(1)}\left( \frac{d}{dt} H_M(\sig,\alpha;t)
\right) \, .
\end{equation}
The term $\frac{d}{dt} H_M(\sig,\alpha;t)$ includes Gaussian
variables, and one may apply to it the
generalized Wick's formula (Gaussian integration-by-parts) for
correlated Gaussian variables, $x_1,\dots,x_n$:
\begin{multline}
\langle x_1\, \psi(x_1,\dots,x_n) \rangle \ = \\
= \ \sum_{j=1}^n \langle x_1 x_j \rangle\,
\langle \frac{\partial \psi(x_1,\dots,x_n)}{\partial x_j} \rangle \, .
\end{multline}
The result is:
\begin{equation}
- \frac{\beta}{M} \E_t^{(1)}\left( \frac{d}{dt} H_M(\sig,\alpha;t)
\right)\ =
\ \frac{\beta^2}{4}\, \E_t^{(2)}\left(
\varphi \right) \,
\end{equation}
where
\begin{multline}
\varphi(\sig,\alpha;\sig',\alpha')\ =\\
= \ \frac{d}{dt}
\Ev{(H_M(\sig,\alpha;t) - H_M(\sig',\alpha';t))^2 \Big\vert
\tilde{q}_{\alpha,\alpha'}, q_{\sig,\sig' }} \\
= \ [f(q_{\sig,\sig'}) - f(\tilde{q}_{\alpha,\alpha'})]
- (q_{\sig,\sig'} - \tilde{q}_{\alpha,\alpha'})
f'(\tilde{q}_{\alpha,\alpha'}) \, .
\end{multline}
Therefore,
\begin{eqnarray}
\frac{d}{dt}P_M(\beta,h;t) \ = \ \frac{\beta^2}{4} \ \times \
\qquad \qquad \qquad \qquad \qquad \qquad \\
\Ert{[f(q_{\sig,\sig'}) - f(\tilde{q}_{\alpha,\alpha'})]
- (q_{\sig,\sig'} - \tilde{q}_{\alpha,\alpha'})
f'(\tilde{q}_{\alpha,\alpha'}) } \ge 0 \nonumber \, .
\end{eqnarray}
The last inequality, which is crucial for us,
follows from the assumed convexity of $f$.
For the SK model, the above expression simplifies to
$\Ert{
(q_{\sig,\sig'}-\tilde{q}_{\alpha, \alpha'} )^{2} } $.
Putting the positivity of the derivative together with
(\ref{eq:int0}) and (\ref{eq:int1})
clearly implies the first bound in (\ref{eq:AFN}).
As was noted earlier, a particular class of random overlap
structures is provided by the Derrida-Ruelle probability
cascade models (GREM) of~\cite{Ru}, which are parametrized
by a monotone function $x: [0,1] \to [0,1]$. These models have
two nice features: {\em i.} the distribution of
$\{\xi_{\alpha}\}$ is invariant, except for a deterministic scaling
factor, under the multiplication by random factors as in
(\ref{ew:G}) (consequently the value of $G_M(\ldots,\mu_{x(\cdot)})$
for such ROSt does not depend on $M$),
{\em ii.} quantities like $G_M(\ldots,\mu_{x(\cdot)})$
can be expressed as the boundary values of the solution of a
certain differential equation, which depends on $x(\cdot)$.
Evaluated for such models $G_M(\ldots,\mu_{x(\cdot)})$ reproduces
the Parisi functional for each value of the order parameter
$x(\cdot)$. The Parisi solution is obtained by optimizing (taking
the {\em inf}) over the order parameter $x(\cdot)$. This relation
gives rise to the second inequality in (\ref{eq:AFN}).
{\em ii.} To prove (\ref{eq:BFN}) we need to supplement
the first inequality in (\ref{eq:AFN}) by an opposite bound.
Our analysis is streamlined by continuity arguments,
which are enabled by the following basic estimate (proven by
two elementary applications of the Jensen inequality).
\noindent{\bf Lemma 2} {\em
Let $Z(H)$ denote the partition function for a system with
the Hamiltonian $H(\sigma)$, and let $U(\sigma)$ be, for
each $\sigma$, a centered Gaussian variable which is
independent of $H$. Then
\begin{equation} \label{Pb}
0 \ \le \ \E\left(\log \frac{Z(H+U)}{Z(H)} \right) \ \le \
\frac{1}{2}\, \E(\, U^2 \, ) \, .
\end{equation}
}
Using the above, it suffices
to derive our result for interactions with the sum over $r$, in
(\ref{eq:pspin}), truncated at some finite value.
A convenient tool is provided by the superadditivity of
$Q_N \equiv N\, P_N$, which was established in the work of
Guerra-Toninelli~\cite{GT} and its extensions~\cite{GT2,CDGG}.
The statement is that for the systems discussed here (and in fact
a broader class) for each $M, N \in \N$
\begin{equation}
\label{eq:superadd}
Q_{M+N}(\beta,h) \geq
Q_M(\beta,h) + Q_N(\beta,h)\, .
\end{equation}
The superadditivity was used in \cite{GT} to establish the existence
of the limit $\lim_{N\to \infty} P_N$. However, it has a
further implication based on the following useful fact.
\noindent{\bf Lemma 3 }
{\em
For any superadditive sequence $\{Q_N\}$
satisfying (\ref{eq:superadd}) the following limits exist and satisfy
\begin{equation}
\lim_{N \to \infty} \, Q_N / N
= \lim_{M \to \infty} \liminf_{N \to \infty}\, \,
[Q_{M+N} - Q_N]/M \, .
\end{equation}
}
For our purposes, this yields:
\begin{equation}
\lim_{N \to \infty} P_N \ = \
\lim_{M \to \infty} \liminf_{N \to \infty}\,
\frac{1}{M} \, \E\left(\log \frac{Z_{N+M}}{Z_{N}} \right) \, .
\end{equation}
We now claim, based on an argument employing the
cavity picture, that for any $M$
\begin{multline}
\label{eq:Opp}
\liminf_{N \to \infty}
\frac{1}{M} \E\left(\log \frac{Z_{N+M}}{Z_{N}} \right)
\ \ge \ \inf_{(\Omega,\mu)} G_M(\beta,h;\mu)
\, ,
\end{multline}
which would clearly imply (\ref{eq:BFN}).
The reason for this inequality is that when a block of
$M$ spins is added to a much larger ``reservoir'' of $N$ spins,
the change in the free energy is
exactly in the form of (\ref{ew:G}) -- except for corrections
whose total contribution to $G_M$ is of order $O(\frac{M}{N})$.
% (due to the
(The spin-spin couplings within the
smaller block and the subleading terms from the change
$N \mapsto (N+M)$ in (\ref{eq:pspin}).)
Thus, the larger block of spins acts as a ROSt.
To see that in detail, let us split the system of $M+N$ spins
into $\sigma = ( \tilde{\sigma}, \alpha)$, with
$\tilde{\sigma}= (\sigma_1,\dots,\sigma_M )$ and $\alpha =
(\sigma_{M+1},\dots,\sigma_{M+N} )$. With this notation, the
interaction decomposes into
\begin{equation}
K_{M+N}(\sig) \ = \
\widetilde{K}_{N}(\alpha)
+ \sum_{j=1}^M \widetilde{\eta}_{j,\alpha} \, \sigma_j
+U(\tilde{\sigma}, \alpha)
\end{equation}
where: {\em i.} $\{\widetilde{K}_N(\alpha) \}$ consists of the
terms of $K_{M+N}(\sig) $ which involve only spins in the larger block,
{\em ii.} the second summand includes all the terms which involve exactly
one spin in the smaller block, and {\em iii.}
$U$ consists of the remaining terms of $K_{M+N}(\sig) $, including the
spin-spin interactions within the smaller block.
One should note that
$\{\widetilde{K}_N(\alpha) \} \neq \{ K_N(\alpha) \}$ since,
as a consequence of the addition of the smaller block, the terms
in $\{\widetilde{K}_N(\alpha) \}$ are weighted by powers of $(N+M)$
rather than $N$, as presented in (\ref{eq:pspin}). By the law of
addition of independent Gaussian variables, $ \{ K_N(\alpha) \}$
(which are of higher variance than $ \{ \widetilde{K}_N(\alpha) \}$ )
have the same distribution as the sum of independent variables
\begin{equation}
\left\{ \widetilde{K}_N( \alpha) + \sqrt{ \frac{M}{2}}\kappa_{\alpha}
\right\} \, ,
\end{equation}
where $\{ \kappa_{\alpha} \}$ are centered Gaussian
variables independent of $\widetilde{K}_N( \alpha) $.
Up to factors $[1+O( \frac{M}{N})]$, the covariances of
$\{\widetilde{\eta}_{j, \alpha} \}$ and $\{ \kappa_{\alpha} \}$
satisfy (\ref{eq:etacov}) and (\ref{eq:kapcov}), respectively,
and
\begin{equation} \label{Ub}
\frac{1}{M}\, \E ( U( \tilde{\sigma}, \alpha)^2 ) \le C \frac{M}{N} \, .
\end{equation}
Taking
\begin{equation}
\xi_{\alpha} := \mbox{ exp} \left[ \beta \left( \widetilde{K}_N(\alpha)+ h
\sum_{i=1}^{N} \alpha_i \right) \right],
\end{equation}
we find that (\ref{eq:Opp}) follows by directly substituting the above into
(\ref{ew:G}) (using (\ref{Pb}) and (\ref{Ub})).
\qed
\noindent{\em Discussion \/ }
At first glance, the recent result of \cite{Gue} may be
read as offering some support to the widely shared belief that the
Parisi ansatz has indeed provided the solution of the SK model.
However, we showed here that the Guerra bound is part of a broader
variational principle in which no reference is made to the
key assumption of \cite{P} that in the limit $N\to \infty$
the SK Gibbs state develops a hierarchal organization.
The reasons for such an organization,
which is equivalently expressed in terms of
``ultrametricity'' in the overlaps $q_{\sig,\sig'}$,
are not a-priori obvious.
(A step, approaching the issue from a dynamical
perspective, was taken in ref~\cite{RA}, but this result has
yet to be extended to the interactive cavity evolution.)
Our result (\ref{eq:AFN}) raises the possibility
that perhaps some other organizing principles may lead to
even lower upper-bounds.
This reinstates
the question whether the ultrametricity
assumption, which has enabled the calculation of \cite{P},
is correct in the context of the SK-type models.
It should be emphasized, however, that the question is not whether
the SK model exhibits replica symmetry breaking at low temperatures.
That, as well as many other aspects of the accepted
picture, are supported by both intuition and by rigorous
results (\cite{ALR,FZ,PS,Tal,NS}).
The question concerns the validity of a solution-facilitating
ansatz about the hierarchal form of the replica symmetry breaking.
The interest in this question is enhanced by fact that this
assumption yields a computational tool
with many other applications~\cite{MPV}.
\noindent{\em Acknowledgements \/ } This work was supported, in part,
by NSF grant PHY-9971149. R. Sims and S. Starr gratefully
acknowledge the support of NSF Postdoctoral Fellowships (MSPRF).
% \bibliography{SK} % to rerun bibtex
% \bibliographystyle{apsrev}
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\begin{thebibliography}{15}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
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\def\bibnamefont#1{#1}\fi
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\def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
\def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
\def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}
\bibitem[{\citenamefont{Sherrington and Kirkpatrick}(1975)}]{SK}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Sherrington}} \bibnamefont{and}
\bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Kirkpatrick}},
\bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{35}},
\bibinfo{pages}{1792} (\bibinfo{year}{1975}).
\bibitem[{\citenamefont{Parisi}(1980)}]{P}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Parisi}}, \bibinfo{journal}{J.
Phys. A: Math. Gen.} \textbf{\bibinfo{volume}{13}}, \bibinfo{pages}{1101}
(\bibinfo{year}{1980}).
\bibitem[{\citenamefont{Mezard et~al.}(1987)\citenamefont{Mezard, Parisi, and
Virasoro}}]{MPV}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Mezard}},
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Parisi}}, \bibnamefont{and}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Virasoro}},
\emph{\bibinfo{title}{Spin glass theory and beyond}},
vol.~\bibinfo{volume}{9} of \emph{\bibinfo{series}{World Scientific Lecture
Notes in Physics}} (\bibinfo{publisher}{World Scientific Publishing Co.,
Inc.}, \bibinfo{address}{Teaneck, NJ}, \bibinfo{year}{1987}).
\bibitem[{\citenamefont{Guerra}(2003)}]{Gue}
\bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Guerra}},
\bibinfo{journal}{Comm. Math. Phys.} \textbf{\bibinfo{volume}{233}},
\bibinfo{pages}{1} (\bibinfo{year}{2003}).
\bibitem[{\citenamefont{Ruelle}(1987)}]{Ru}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Ruelle}},
\bibinfo{journal}{Comm. Math. Phys.} \textbf{\bibinfo{volume}{108}},
\bibinfo{pages}{225} (\bibinfo{year}{1987}).
\bibitem[{\citenamefont{Derrida}(1981)}]{Der}
\bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Derrida}},
\bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{24}},
\bibinfo{pages}{2613} (\bibinfo{year}{1981}).
\bibitem[{\citenamefont{Guerra and Toninelli}()}]{GT2}
\bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Guerra}} \bibnamefont{and}
\bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Toninelli}},
\bibinfo{note}{preprint}, \eprint{cond-mat/0208579}.
\bibitem[{\citenamefont{Guerra and Toninelli}(2002)}]{GT}
\bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Guerra}} \bibnamefont{and}
\bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Toninelli}},
\bibinfo{journal}{Comm. Math. Phys.} \textbf{\bibinfo{volume}{230}},
\bibinfo{pages}{71} (\bibinfo{year}{2002}).
\bibitem[{\citenamefont{Contucci et~al.}(2003)\citenamefont{Contucci,
Degli~Esposti, Giardina, and Graffi}}]{CDGG}
\bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Contucci}},
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Degli~Esposti}},
\bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Giardina}}, \bibnamefont{and}
\bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Graffi}},
\bibinfo{journal}{Comm. Math. Phys.} \textbf{\bibinfo{volume}{236}},
\bibinfo{pages}{55} (\bibinfo{year}{2003}).
\bibitem[{\citenamefont{Ruzmaikina and Aizenman}(2003)}]{RA}
\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Ruzmaikina}} \bibnamefont{and}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Aizenman}}
(\bibinfo{year}{2003}), \bibinfo{note}{preprint}.
\bibitem[{\citenamefont{Aizenman et~al.}(1987)\citenamefont{Aizenman, Lebowitz,
and Ruelle}}]{ALR}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Aizenman}},
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Lebowitz}}, \bibnamefont{and}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Ruelle}},
\bibinfo{journal}{Comm. Math. Phys.} \textbf{\bibinfo{volume}{112}},
\bibinfo{pages}{3} (\bibinfo{year}{1987}).
\bibitem[{\citenamefont{Fr{\"o}hlich and Zegarlinski}(1987)}]{FZ}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Fr{\"o}hlich}}
\bibnamefont{and}
\bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Zegarlinski}},
\bibinfo{journal}{Comm. Math. Phys.} \textbf{\bibinfo{volume}{112}},
\bibinfo{pages}{553} (\bibinfo{year}{1987}).
\bibitem[{\citenamefont{Pastur and Shcherbina}(1991)}]{PS}
\bibinfo{author}{\bibfnamefont{L.~A.} \bibnamefont{Pastur}} \bibnamefont{and}
\bibinfo{author}{\bibfnamefont{M.~V.} \bibnamefont{Shcherbina}},
\bibinfo{journal}{J. Statist. Phys.} \textbf{\bibinfo{volume}{62}},
\bibinfo{pages}{1} (\bibinfo{year}{1991}).
\bibitem[{\citenamefont{Talagrand}(2003)}]{Tal}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Talagrand}},
\emph{\bibinfo{title}{Spin glasses : A challenge for mathematicians. Mean
field theory and Cavity method}} (\bibinfo{publisher}{Springer Verlag},
\bibinfo{address}{Berlin}, \bibinfo{year}{2003}), \bibinfo{note}{to appear}.
\bibitem[{\citenamefont{Newman and Stein}(2003)}]{NS}
\bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Newman}} \bibnamefont{and}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Stein}},
\bibinfo{note}{preprint}, \eprint{cond-mat/0301202}.
\end{thebibliography}
\end{document}
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