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\begin{document}
\title{General properties of overlap probability distributions in
disordered spin systems. \\Toward Parisi ultrametricity.}
\date{}
\author{Stefano Ghirlanda\small{*} \and Francesco Guerra\small{**}}
\maketitle
\centerline{\small{* Zoologiska Institutionen, Stockholms Universtitet, S-106 91 Stockholm, Sweden}}
\centerline{\small{** Dipartimento di Fisica, Universit\`a di Roma ``La Sapienza'', I-00185 Roma, Italy}}
\centerline{\small{\& INFN, Sezione di Roma 1.}}
\begin{abstract}
For a very general class of probability distributions in disordered
Ising spin systems, in the thermodynamical limit, we prove the following
property for overlaps among real replicas. Consider the overlaps among $s$ replicas. Add one replica $s+1$. Then, the overlap $q_{a,s+1}$ between one of the first $s$ replicas, let us say $a$, and the added $s+1$ is either independent of the former ones, or it is identical to one of the overlaps $q_{a b}$, with $b$ running among the first $s$ replicas, excluding $a$. Each of these cases has equal probability $1/s$.
\end{abstract}
\section{Introduction}
In this paper we focus on general properties of overlap distributions
in statistical-mechanical models made up of Ising spins (see below for
definitions). Historically,
these properties have been considered for the first time in spin-glass
models \cite{spi87}, so that for convenience we take them as a starting point in
our discussion, and generalise our results later on.
The problem of finding the phase
structure of short-ranged models for spin glasses has proved extremely
difficult, and yet remains unsolved. An important result, though, has
been achieved with Parisi's solution of the Sherrring\-ton-Kirkpatrick
(SK) model (a mean-field
approximation to more realistic ones), whose
hamiltonian we recall:
\begin{equation}\label{HSK}
{\mathcal H}_J\{\sigma\} = -\unsu{\sqrt{N}}\sum_{(ik)} J_{ik}
\sigma_i \sigma_k\,.
\end{equation}
The $\sigma_i$'s ($i=1,\ldots,N$) are Ising spins and the $J_{ik}$'s
(collectively noted $J$) are random variables drawn from
independent unit normal distributions, with the constraints $J_{ik}=J_{ki}$ and
$J_{ii}=0$.
The sum runs over all couples $(ik)$, with $1\le iFrom \mref{Es+1s+2} we can derive other known ultrametric equalities,
for example:
\begin{equation}
\me{q_{12}^2q_{34}^2}=\unte\me{q_{12}^4}+\frac{2}{3}\me{q_{12}^2}^2\,,
\end{equation}
again in agreement with \mref{ro1234} obtained from Parisi's solution.
In the following section we generalise formulae \mref{Eas+1} and
\mref{Es+1s+2} to arbitrary (integer) powers of overlaps.
\section{Auxiliary interactions and overlap probability distributions}
Let us consider the SK model in the presence of an external field:
\begin{eqnarray}\nonumber
A_{J,J'}\{\sigma\}&\defeq&
A_J\{\sigma\}+\frac{\lambda}{N}\sum_{i=1}^{N}J_i'
\sigma_i \\ &\defeq& A_J\{\sigma\} +\lambda I_{J'}\{\sigma\}\,,
\end{eqnarray}
where the random variables $J_i'$ are independt from the $J_{ik}$'s
and with the same distribution. We assume that $\lambda$ is ``small'',
since in the end we will take it to zero to recover the free SK model.
Theorem \mref{fluctAF} can be generalised to the present case, since
it only relies on self-averaging of the internal energy:
\begin{equation}
\lim_{N\rightarrow\infty}\left(\me{A_{J,J'}\{\sigma^a\}F_s(q)}-\me{A_{J,J'}\{\sigma\}}\me{F_s(q)}\right)=0\,,
\end{equation}
where now $\me{\cdot}$ implies averaging over the $J'$ variables as
well. Using the same procedure as in the preceding section, but now
integrating and deriving with respect to the $J'$ variables, we get
the completely analogous formula:
\begin{equation}
\cond{q_{a,s+1}}=\unsu{s}\me{q}+\unsu{s}\sum_{b\neq a}q_{ab}\,,\\
\end{equation}
which continues to hold when $\lambda$ is taken to zero (after having
taken the thermodynamical limit). The only difference between this
formula and \mref{Eas+1} is that here overlaps appear at the first
power.
It is now clear that we can consider auxiliary interactions of the
general form:
\begin{equation}\label{moltispin}
\lambda_rI_r\{\sigma\}\defeq \frac{\lambda_r}{N^{(r+1)/2}}\sum_{(i_1\ldots i_r)}
J_{i_1,\ldots,i_r}'\sigma_{i_1}\cdots\sigma_{i_r}\,,
\end{equation}
the former case being $r=1$. So we end up with the formula:
\begin{equation}\label{Er1}
\cond{q_{a,s+1}^r}=\unsu{s}\me{q^r}+\unsu{s}\sum_{b\neq a}q_{ab}^r\,,\\
\end{equation}
valid for the free SK model when $\lambda_r\rightarrow 0$. A similar
formula is valid for $\cond{q_{s+1,s+2}^r}$ as a generalisation of \mref{Es+1s+2}.
We have thus obtained the main result of this paper:
\newline
{\Large\sc theorem.} \emph{Given the overlaps among $s$ real
replicas, the overlap between one of these and an additional replica
is either independent of the former overlaps or it is identical to one of
them, each of these cases having probability $1/s$:}
\begin{equation}\label{result}
\rho_{a,s+1}(q_{a,s+1}|{\mathcal
A}_s)=\unsu{s}\rho(q_{a,s+1})+\unsu{s}\sum_{b\neq
a}\delta(q_{a,s+1}-q_{ab})\,,
\end{equation}
\emph{where $\rho_{a,s+1}(\cdot|{\mathcal A}_s)$ is the conditioned
distribution of $q_{a,s+1}$ given the overlaps in ${\mathcal
A}_s$.}
{\Large\sc proof.} The theorem is proved by \mref{Er1} and the fact that the overlaps are
bounded.
\newline
{\Large\sc corollary:} \emph{The distribution
of $q_{s+1,s+2}$ conditioned to the overlaps in ${\mathcal A}_s$ is
given by:}
\begin{equation}\label{corollary}
\rho_{s+1,s+2}(q_{s+1,s+2}|{\mathcal
A}_s)=\frac{2}{s+1}\rho(q_{s+1,s+2})+\frac{2}{s(s+1)}\sum_{a