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GREM models, Thermodinamical limit
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\begin{document}

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\begin{center}{\sc\Large Thermodynamical Limit for Correlated Gaussian
Random Energy Models}
\end{center}
\begin{center}{ P. Contucci, M. Degli Esposti,
C. Giardin\`a, S. Graffi}\\
{\small Dipartimento di Matematica} \\
    {\small Universit\`a di Bologna,
    40127 Bologna, Italy}\\
    {\small {e-mail:
    $\{$contucci,desposti,giardina,graffi$\}$@dm.unibo.it}}
\end{center}
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\begin{abstract}\noindent
Let  $\{E_{\s}(N)\}_{\s\in\Sigma_N}$ be a family of
$|\Sigma_N|=2^N$ centered unit Gaussian  random variables defined
by the covariance matrix $C_N$ of elements $\displaystyle
 c_N(\s,\tau):=\av{E_{\s}(N)E_{\tau}(N)}$, and $H_N(\s) = -
\sqrt{N} E_{\s}(N)$ the  corresponding random Hamiltonian.  Then
the quenched thermodynamical limit exists if, for every
decomposition $N=N_1+N_2$, and all pairs $(\s,\t)\in
\Sigma_N\times \Sigma_N$:
$$
c_N(\s,\tau)\leq \frac{N_1}{N}\;c_{N_1}(\pi_1(\s),\pi_1(\tau))+
\frac{N_2}{N}\;c_{N_2}(\pi_2(\s),\pi_2(\tau))
$$
where $\pi_k(\s), k=1,2$ are the projections of $\s\in\Sigma_N$
into $\Sigma_{N_k}$. The condition is explicitly verified for the
Sherrington-Kirckpatrick,  the even $p$-spin, the Derrida REM and
the Derrida-Gardner GREM models.
\end{abstract}
\vspace{0.5cm}\noindent
\section{Introduction, Definitions and Results}
It has recently been proved by Guerra and Toninelli \cite{GuTo}
that for the Sherrington-Kirckpatrick (hereafter SK) model (as
well as for the even-$p$-spin models) the thermodynamical limit
exists for the quenched free energy and almost everywhere for its
random realizations. In this paper we single out general
sufficient conditions that imply the existence of the quenched
thermodynamical limit for any correlated Gaussian random energy
model. Our analysis thus includes as special cases not only the
even $p$ spin models (in particular the SK one, $p=2$) but also
the Derrida REM model\cite{De1},\cite{De2} and the Derrida-Gardner
GREM\cite{DeGa}.

The paper is organized as follows: in this section we introduce
the definitions and state the results. In section 3, after
introducing and elucidating the operation of {\it lifting} for a
family of Gaussian random variables, we describe the proof of our
theorem. In section 4 we show how our analysis can be applied to
the specific examples listed above.

To define the set up we consider a disordered model having $2^N$
energy levels where $N$ is the size of the system. We label the
energy levels by the index $\s = \{\s_1,\s_2,\ldots,\s_N\}$ where
each $\s_i$ takes the values $\pm 1$ for $i=1,\ldots,N$. We denote
$\Sigma_N$ the set of all $\s$. Then $|\Sigma_N|=2^N$. Clearly
$\Sigma_N$ coincides with the space of all the possible $2^N$
Ising configurations of length $N$.
\begin{definition}
\label{modello} Denote $\{E_{\s}(N)\}_{\s\in\Sigma_N}$  a family
of $2^N$ {\em centered unit Gaussian} random variables: \be
\label{centered} \av{E_{\s}(N)} = 0 \; , \ee and covariance matrix
$C_N$ with elements defined by
\begin{eqnarray}
\label{unit} c_N(\s,\s) := \av{E^2_{\s}(N)} = 1 \; ,
\\
 c_N(\s,\t) := \av{E_{\s}(N)E_{\tau}(N)} \; .
\end{eqnarray}
 Here $\av{-}$
denotes expectation with respect to the probability measure \be
dP\left(E_1,\ldots,E_{2^{N}}\right)
\,=\,\frac{1}{\sqrt{(2\pi)^{2^N}\dete{C}}}\,\,\,
e^{-\frac{1}{2}\<E,\,C^{-1}E\>}\,\,dE_1\cdots dE_{2^{N}} . \ee
\end{definition}

\begin{definition}
\par\noindent
\begin{enumerate}
\item
For each $N$ the Hamiltonian is given by \be H_N(\s) =
-\sqrt{N}E_{\s}(N) \; . \label{ee} \ee
\item
The partition function of the system is: \be \label{part}
Z_N(\b,E) = \sum_{\s} e^{-\b H_N(\s)} = \sum_{\s} e^{\b\sqrt{N}
E_{\s}(N)} \ee
\item
The  {\em quenched} free energy $f_N(\b)$ of the system is defined
as: \be -\b f_N(\b) := \a_N(\b) := \frac{1}{N}\; \av{\ln
Z_N(\b,E)} \; . \label{fe} \ee
\end{enumerate}
\end{definition}
\begin{remark}
{\rm From now we write $ E_{\s}(N)= E_{\s}$, dropping the
$N$-dependence. Remark moreover that the above definition includes
Gaussian families of the form
\begin{eqnarray}
E_\s(N) \, = \, J_0 + \sum_{i}J_i\s_i + \sum_{i,j}J_{i,j}\s_i\s_j
+ \sum_{i,j,k}J_{i,j,k}\s_i\s_j\s_k + \nonumber
\\
+\ldots
+\sum_{i_1,i_2,...,i_N}J_{i_1,i_2,...,i_N}\s_{i_1}\s_{i_2}...\s_{i_N}
\; \label{genercrem}
\end{eqnarray}
in which every $J$ is an indipendent Gaussian variable. }
\end{remark}
{\bf Examples}.
\begin{enumerate}
\item The SK model.
Consider first the model defined by \be \label{SKN}
 E_{\s}
:=\frac{1}{N}\sum_{i,j=1}^N J_{i,j}\s_i\s_j \ee where the
$J_{i,j}$ are $N^2$ i.i.d. unit Gaussian random variables. A short
computation yields
$$
{\rm Av}(E_{\s}E_{\t})=[q_N(\s,\t)]^2
$$
where, as usual \be \label{overlap}
q_N(\s,\t):=\frac{1}{N}\sum_{k=1}^N \s_k\t_k \label{chiu} \ee is
the overlap between the $\s$ and $\t$ spin configurations. The
standard SK model is instead defined by \be \label{SK}
 E_{\s}^{SK}
:=\frac{1}{N}\sum_{i<j=1}^N J_{i,j}\s_i\s_j \; . \ee However the
quenched free energy densities (\ref{fe}) of the two models
coincide up to a rescaling of the temperature, i.e.: \be
\a_N^{SK}(\sqrt{2}\beta) \, = \, \a_N (\b) \, , \label{id} \ee In
fact,
 $J_{i,j}\s_i\s_j$ are centered, unit and i.i.d. Gaussian
random variables $\forall\,(i,j)$, and $J_{i,j}\s_i\s_j=
J_{j,i}\s_j\s_i$. Hence $\displaystyle
J_{i,j}\s_i\s_j+J_{j,i}\s_j\s_i\, \mathrel{\mathop{=}\limits^{\cal
D}}\,\sqrt{2}J_{i,j}\s_i\s_j$ (here $\displaystyle
\mathrel{\mathop{=}\limits^{\cal D}}$ denotes equality in
distribution of two random variables).  Therefore, taking into
account also the $N$ diagonal terms: \be \sqrt{N}E_{\s} \,
\mathrel{\mathop{=}\limits^{\cal D}}\, \sqrt{N}\sqrt{2}E^{SK}_{\s}
+ J \; , \label{lkj} \ee where $J$ is a centered unit Gaussian
variable. By (\ref{part},\ref{fe}) formula (\ref{lkj}) immediately
yields the relation (\ref{id}).
\item
The $p$-spin models. Here we consider the model: \be \label{SKp}
E_{\s}:=\sqrt{\frac{1}{N^p}}\sum_{i_1,\ldots,i_p=1}^NJ_{i_1,\ldots,i_p}\s_{i
_1 }\cdots\s_{i_p} \ee where the $J_{i_1,\ldots,i_p}$ are once
more i.i.d. unit Gaussian random variables. As before, a short
computation yields \be {\rm Av(E_{\s}E_{\t})}=[q_N(\s,\t)]^p \ee
\item
The Derrida REM. Here the model is specified by Definition
\ref{modello} with \be {\rm Av}(E_{\s}E_{\t})=\delta(\s,\t) \ee
\item The Derrida-Gardner GREM. Its inclusion into the above framework is
described in detail in Section 3.3.
\end{enumerate}
\begin{definition}
\label{def:pai} For each $\s\in\Sigma_N$ let $\pi_1$ and $\pi_2$
be the two canonical projections over the two  subsets
$\Sigma_{N_1}$ and $\Sigma_{N_2}$, generated by a partition ${\cal
P}$ of the coordinates $(\s_1,\ldots,\s_N)$ into a subset of $N_1$
coordinates and into a complementary set of $N_2$ coordinates:
$N_1+N_2=N$, $\Sigma_N=\Sigma_{N_1}\times\Sigma_{N_2}$,
$\pi_1\otimes\pi_2=1_{\Sigma_N}$.
\end{definition}
\noindent (Example:  $N=4$;  $\sigma\in \Sigma_4$ with coordinates
denoted $\{\sigma_1, \s_2, \s_3, \s_4 \}$. Consider for
$N_1=N_2=2$ the partition
 ${\cal P}\s=(\s_1,\s_2)\cup (\s_3,\s_4)$. Then
$\Sigma_N=\Sigma_{N_1}\times\Sigma_{N_2}$ and the two projections
$\pi_k:\Sigma_N\to \Sigma_{N_k}$, $k=1,2$ act in the following
way: $\pi_1(\sigma_1, \s_2, \s_3, \s_4) = (\sigma_1, \s_2)$ and
$\pi_2(\sigma_1, \s_2, \s_3, \s_4) = (\sigma_3, \s_4)$).
\par\noindent
Our main result is the following:
\begin{theorem}
\label{th:gnocca} Let the covariance matrices $C_N$ fulfill the
condition: \be c_N(\s,\t) \,-
\,\frac{N_1}{N}\;c_{N_1}(\pi_{1}(\s),\pi_{1}(\t)) \, - \,
                 \frac{N_2}{N}\;c_{N_2}(\pi_{2}(\s),\pi_{2}(\t)) \; \le
\; 0 \; , \label{punct} \ee for every $N\ge \tilde{N}$, every
$(\s,\t)\in \Sigma_N\times \Sigma_N$ and every decomposition
$N_1+N_2=N$. Then the thermodynamical limit exists, in the sense
that \be \lim_{N\to\infty}\frac{1}{N}{\rm Av}(\log Z_N(\beta))
=\sup_{N}\frac{1}{N}{\rm Av}(\log Z_N(\beta)) \; . \label{gnocca}
\ee
\end{theorem}
\begin{remark}
{\rm The  result $(\ref{gnocca})$ can be extended to the
almost-everywhere convergence of free energy density, internal
energy and ground state energy with elementary probability methods
(see $\cite{GuTo}$)}
\end{remark}
\begin{remark}
{\rm The conditions (\ref{punct}) are not }necessary. {\rm The
proof itself will show that we only need the sign of the quantity
in the left hand side of (\ref{punct}) in average, not pointwise.
Moreover the condition (\ref{centered}) can be replaced by a more
general small deviation vanishing for large $N$ and (\ref{unit})
by a uniform (in N) bound over the diagonal terms. We plan to
return over such a general case elsewhere.}
\end{remark}
\begin{remark}
{\rm It is still an open interesting question whether the class of
models
 we control the thermodynamical limit of do have, in that
limit, the properties axiomatically introduced by Ruelle in
$\cite{rugrem}$ to define directly the infinite particle systems.
To this purpose see $\cite{BS}$, $\cite{BK1}$, $\cite{BK2}$ and
$\cite{BK3}$.}
\end{remark}

\section{Proof}
Within this section it is useful to consider $2$ identical {\it
copies} of the same system: the system $1$ is assigned the
Hamiltonian $H(\s)$ and the system $2$ the Hamiltonian $H(\t)$.

\vspace{0.2cm} \noindent
\begin{definition}
The {\it quenched} measure over the two copies $\<-\>$ is defined
by \be <-> \, = \, {\rm Av}{[Z(\beta,E)]^{-2}\sum_{(\s,\t)\in
\Sigma_N\times \Sigma_N} - \; e^{\beta (H(\s)+H(\t))}} \, .
\label{bracket} \ee The definition may of course be generalized to
$r$ copies.
\end{definition}

We want now to embed a Gaussian system $\{E_\s\}_{\Sigma_K}$ into
a larger one $\{E_\t\}_{\Sigma_L}$ for some $K<L$. In particular
we want to embed two of them of size $N_1$ and $N_2$ into one of
size $N=N_1+N_2$. Our embedding procedure is defined in terms of
the two canonical projections $\pi_j$, $j=1,2$ from $\Sigma_N$ to
$\Sigma_{N_j}$ given in Definition (\ref{def:pai}).
\begin{definition}
Given the family $\{E_\mu\}_{\Sigma_{N_1}}$ of size $N_1$ we {\rm
lift} it to one of size $N$: $\{E^{(1)}_\s\}_{\Sigma_{N}}$
defining \be E^{(1)}_{\s} \, \mathrel{\mathop{=}\limits^{\cal D}}
\, E_{\pi_1(\s)} \; . \ee Moreover starting from
$\{E_\mu\}_{\Sigma_{N_2}}$ we define in the same way
$\{E^{(2)}_\s\}_{\Sigma_{N}}$ by \be E^{(2)}_{\s} \,
\mathrel{\mathop{=}\limits^{\cal D}} \, E_{\pi_2(\s)} \; . \ee
Having defined each family $\{E_\s\}_{\Sigma_{N}}$,
$\{E^{(1)}_\s\}_{\Sigma_{N_1}}$ and
$\{E^{(2)}_\s\}_{\Sigma_{N_2}}$ we specify their joint
distribution requiring mutual independence.
\end{definition}
\begin{remark}:
{\rm The embedded Gaussian systems $\{E^{(1)}_\s\}_{\Sigma_{N_1}}$
and $\{E^{(2)}_\s\}_{\Sigma_{N_2}}$ are degenerate: In fact for
all $\s$ and $\t$ such that $\pi_1(\s) = \pi_1(\t)$ \be E^{(1)}_\s
\, = \, E^{(1)}_\t \; . \ee Summarizing we define the joint
measure of $\{E_\s\}_{\Sigma_{N}}$,
$\{E^{(1)}_\s\}_{\Sigma_{N_1}}$ and
$\{E^{(2)}_\s\}_{\Sigma_{N_2}}$ $d\hat{P}=dP dP_{1} dP_{2}$
defined by the three covariances $C_{N}$, $C_{N_1}$ and
$C_{N_2}$.}
\end{remark}
{\bf Proof of THEOREM 1.} \\
We proceed in three steps. \\
\noindent
%\vskip .3truecm
0) \hskip .3truecm {\bf Interpolation}
\par\noindent
Given a pair $(\pi_1,\pi_2)$ as before,  following \cite{GuTo}, we
 pick three {\it independent} Gaussian systems
$E^{(j)}_{\pi_j(\sigma)}$, $j=0,1,2$ and introduce the quantity
($\pi_0(\sigma)=\sigma$)

\be \label{zt} H_{(N,N_1,N_2)}(\sigma,t) \; := \;
-\sum_{j=0}^2\sqrt{t_jN_j} E_{\pi_j(\sigma)}^{(j)} \ee where
$t_0=t$ and $t_1=t_2=(1-t)$, and the correspondent partition sum
\be \label{zt2} Z_N(t,\beta) \;: = \; \sum_{\sigma\in\Sigma_N}
e^{-\beta H_{(N,N_1,N_2)}(\sigma,t)}. \ee It is now easy to see
that: \be
 Z_N(1,\beta)=Z_N{(\beta)} \; ,
\label{int1} \ee and
\begin{eqnarray}
Z_N(0,\beta)&=&\sum_{\sigma\in\Sigma_N} e^{\beta (
\sqrt{N_1}E_{\pi_1(\sigma)}^{(1)} \,+
\,\sqrt{N_2}E_{\pi_2(\sigma)}^{(2)})}\nonumber\\
&=&\sum_{\tau\in\Sigma_{N_2}}\,\,\sum_{\sigma\in\Sigma_N;
\,\,\pi_2(\sigma)=\tau} e^{\beta
(\sqrt{N_1}E_{\pi_1(\sigma)}^{(1)} \,+
\,\sqrt{N_2}E_{\tau}^{(2)})}\nonumber\\
&=&\sum_{\tau\in\Sigma_{N_2}}\,e^{\beta
\sqrt{N_2}E_{\tau}^{(2)}}\,\sum_{\g\in\Sigma_{N_1}}
e^{\beta \sqrt{N_1}E_{\g}^{(1)}}\nonumber\\
&=&Z_{N_1}(\beta)\cdot Z_{N_2}(\beta)\label{interpolo}
\end{eqnarray}
\noindent
1) \hskip .3truecm {\bf Boundedness}\\
The Jensen inequality \be \av{\log Z} \, \le \, \log(\av{Z}) \ee
implies \be \frac{1}{N}\av{\log Z_N(\beta)} \; \le \;
\log(2)+\frac{\beta^2}{2} \ee because by (\ref{part})
$\displaystyle \av{Z}=2e^{\b^2/2}$ after performing the Gaussian
integration.
 \noindent
2) \hskip .3truecm {\bf Monotonicity}\\Taking the $t$ derivative
of the logarithm of (\ref{zt2}) we get: (here we abbreviate
$H_{N,N_1,N_2}=H$)

\be \frac{d}{dt}\log
Z_N(t)\,=\,\frac{\beta}{Z_N(t)}\,\sum_{\sigma\in\Sigma_N}\left(\sum_{k=0}^2
\epsilon_k \sqrt{\frac{N_k}{t_k}} E_{\pi_k(\sigma)}^{(k)}
\,e^{-\beta H(\sigma,t)}\right)\, , \ee where $\epsilon_0=1$ and
$\epsilon_1=\epsilon_2=-1$.

We now use the integration by parts formula for correlated
Gaussian variables $\{\xi_i\}$ with covariance $c_{i,j}$, which
states \be \av{\xi_j\cdot f}\,=\,\av{\sum_{k=1}^n
c_{j,k}\cdot\frac{\partial f}{\partial \xi_k}} \; . \ee This
yields
\begin{eqnarray}
\label{sum} \av{\frac{1}{\beta }\frac{d}{dt}\log Z_N(t)}=
\sum_{\sigma\in\Sigma_N}\sum_{k=0}^2\epsilon_k\,\sqrt{\frac{N_k}{t_k}}\,
\av{\frac{E_{\pi_k(\sigma)}^{(k)}\,e^{-\beta H}}{Z_N(t)}}\\
=\sum_{\sigma\in\Sigma_N}\sum_{k=0}^2\epsilon_k\,\sqrt{\frac{N_k}{t_k}}\,
\av{ \sum_{\tau_k\in\Sigma_{N_k}}
c_{N_k}(\pi_k(\sigma),\tau_k)\cdot\frac{\partial}{\partial
E_{\tau_k}^{(k)}}\frac{e^{-\beta H}}{Z_N(t)}}\nonumber
\end{eqnarray}

\noindent Given now $\tau_k \in\Sigma_{N_k}$ fixed, we calculate

\begin{eqnarray}
&&\frac{\partial}{\partial E_{\tau_k}^{(k)}}\frac{e^{-\beta
H(\sigma,t)}}{Z_N(t)}  =
\beta\frac{\sqrt{N_kt_k}\;\delta_{\tau_k}^{\pi_k(\sigma)}\;e^{-\beta
H(\sigma,t)}\cdot Z_N(t)-e^{-\beta H(\sigma,t)}\cdot\frac{\partial
Z_N}{\partial
E_{\tau_k}^{(k)}}}{Z_N^2(t)} \nonumber \\
\,\nonumber\\ & = &
\beta\frac{\sqrt{N_kt_k}\;\delta_{\tau_k}^{\pi_k(\sigma)}
e^{-\beta H(\sigma,t)}\cdot Z_N(t)-\sqrt{N_kt_k}\;e^{-\beta
H(\sigma,t)}\cdot\sum_{\xi\in\Sigma_N,\,\pi_k(\xi)=\tau_k}
e^{-\beta H(\xi,t)}}{Z_N^2(t)}\nonumber
\end{eqnarray}

\noindent The term with $k=0$ in formula (\ref{sum}) is easy to
calculate and we get:
\begin{eqnarray}
&&
N\beta\av{\sum_{\sigma\in\Sigma_N}\sum_{\tau\in\Sigma_N}c_N(\sigma,\tau)
\left[\delta_\tau^\sigma \frac{e^{-\beta H(\sigma,t)}}{Z_N}-
\sum_{\xi\in\Sigma_N} \delta_\xi^\tau e^{-\beta(H(\xi,t)+
H(\sigma,t))}\right]}\,=\,\nonumber\\
 &=& N\beta \av{\sum_{\sigma\in\Sigma_N}
c_N(\sigma,\sigma)\cdot\frac{e^{-\beta H(\sigma,t)}}{Z_N}\;-\;
 \sum_{(\s,\t)\in \Sigma_N\times \Sigma_N}
 c_N(\sigma,\tau)\,e^{-\beta(H(\tau,t)+H(\sigma,t))}}\nonumber\\
 &=& N\beta \brac{1\,-\, c_N(\s,\t)}_t \; ,
\end{eqnarray}
where $<->_t$ is the quenched measure with respect to the
Hamiltonian (\ref{zt}). \vspace{0.3cm} \noindent In the same way
for the term $k=1$ (and similarly for $k=2$) we obtain:
\begin{eqnarray}
N_1\beta\av{\sum_{\sigma\in\Sigma_N}
\sum_{\tau\in\Sigma_{N_1}}c_{N_1}(\pi_1( \sigma),\tau)\left[
\delta^{\tau}_{\pi_1(\sigma)}\frac{e^{-\beta H(\sigma,t)}}{Z_N}-
\sum_{\xi\in\Sigma_N} \delta_{\pi_1(\xi)}^\tau e^{-\beta(H(\xi,t)+
H(\sigma,t))}\right]}\,=\, \nonumber
\\
= N_1\brac{1\,-\, c_{N_1}(\pi_{1}(\s),\pi_1(\t))}_t \; .
\qquad\qquad\qquad\qquad\qquad\qquad\qquad \label{sbbc}
\end{eqnarray}
Summing up the three contributions we obtain:
\begin{eqnarray}
\frac{1}{N}\frac{d}{dt}\av{\log Z_N(t)} = \qquad\qquad\qquad
\qquad\qquad\nonumber
\\
 = -{\beta^2}<c_N(\s,\t) -
\frac{N_1}{N}c_{N_1}(\pi_{1}(\s),\pi_{1}(\t)) -
                 \frac{N_2}{N}c_{N_2}(\pi_{2}(\s),\pi_{2}(\t)) >_t \; ,
\label{der}
\end{eqnarray}
and, by the hypothesis (\ref{punct}): \be \frac{d}{dt}\av{\log
Z_N(t)} \ge 0 \; . \label{pos} \ee Formula (\ref{pos}) together
with the boundary conditions (\ref{int1}) and (\ref{interpolo})
gives for every $N_1+N_2=N$ \be \a_N \ge \frac{N_1}{N}\a_{N_1} +
\frac{N_2}{N}\a_{N_2} \; . \label{sub} \ee This entails Theorem
\ref{th:gnocca} as explained for instance in \cite{Ru2}.
\section{Examples}
\subsection{The SK and even $p$-spin models}
For the sake of completeness we recover here the Guerra-Toninelli
result \cite{GuTo}. First note that by the definition
$(\ref{chiu})$ we have \be q_N(\s,\t) \,-
\,\frac{N_1}{N}q_{N_1}(\pi_{1}(\s),\pi_{1}(\t)) \, - \,
                 \frac{N_2}{N}q_{N_2}(\pi_{2}(\s),\pi_{2}(\t))
\; = \; 0 \; . \label{rfm} \ee so that $(\ref{punct})$ holds as an
equality for $p=1$ (the random field model). By $(\ref{sub})$ this
means that the random field model free energy density doesn't
depend on the size: $\a_N = \a_1$. For $p=2u$ (SK corresponds to
$u=1$) formula $(\ref{rfm})$ together with the convexity of the
function $x\to x^{2u}$ implies  $(\ref{punct})$: \be
q^{2u}_N(\s,\t) \,-
\,\frac{N_1}{N}q^{2u}_{N_1}(\pi_{1}(\s),\pi_{1}(\t)) \, - \,
                 \frac{N_2}{N}q^{2u}_{N_2}(\pi_{2}(\s),\pi_{2}(\t)) \; \le
\; 0 \; . \label{evenp} \ee For the standard $p$-spin model
defined as \be \label{SKp1}
 E_{\s}=\sqrt{\frac{p!}{2N^p}}\sum_{i_1<\ldots <
i_p}J_{i_1,\ldots,i_p}\s_{i_1 }\cdots\s_{i_p} \ee we refer to
\cite{GuTo}
\subsection{The REM}
The model is defined by: \be \av{E_{\s}E_{{\s}'}} =
\delta_{\s,\s'} . \ee Condition $(\ref{punct})$ is verified
because it becomes \be \delta_{\s,\s'} \leq
\frac{N_1}{N}\delta_{\pi_1(\s),\pi_1(\s')} + \frac{N_2}{N}
\delta_{\pi_2(\s),\pi_2(s')} \; . \ee In fact if $\s=\s'$ the
previous formula is an identity. If $\s\neq\s'$ the left hand side
is $0$ but the right hand side is not always zero.  Let us take
for instance $\s = (+,+)$ and $\s'=(+,-)$, $\pi_1(+,+)=+$,
$\pi_1(+-)=+$, $\pi_2(+,+)=+$, $\pi_2(+,-)=-$. In that case the
left hand side is zero and the right hand side is $1/2$.

\subsection{The GREM}
In order to show that our scheme includes the Derrida-Gardner GREM
\cite{DeGa} let first shortly recall its construction and add few
observations. The GREM considers $2^N$ Gaussian random energies
$H(\mu) = \sqrt{N}E_\mu$. Their covariance is given after the
assignment of a {\it rooted tree} with $n$ layers and $2^N$
leaves, $n<N$. The root $\alpha_1^N$-{\it furcates}, the vertices
at the end of the first layer $\alpha_2^N$-{\it furcate}  etc. up
to the vertices at the end of the $n-1$ layer which
$\alpha_n^N$-{\it furcates} on the $2^N$ leaves. The topological
constraint implies $\prod_{i=1}^{n} \alpha_{i}^{N}=2^N$ and the
prime number decomposition theorem imposes that
$\alpha_{i}^{N}=2^{k_i}$ with $k_1 + k_2 + ... k_n = N$. It is
interesting and useful to associate to each leave $\mu$ a
configuration of signs $\{\s_1, \s_2, ..., \s_N\}$. This can be
done observing that the $\alpha_1^N=2^{k_1}$ branches emerging
from the root identify canonically the configurations of $k_1$
spins, the successive branches the configuration of $k_2$ spins
and so on. We have in this way associated to each leaf either a
path (the only one joining the root to it) or a spin
configuration. The model is finally specified by the formula
$E(\mu) = \sum_{i=1}^{n} \epsilon_{i}^{(\mu)}$ where the
$\epsilon_i$ are thrown according to $n$ Gaussians with a ${\rm
Av}(\epsilon_i)=0$ and ${\rm Av}[(\epsilon_i)^2]=a_i$: to each
branch of the tree we associate an independent $\epsilon$ whose
distribution (through its variance) depends only at which layer is
the branch. Defining $\displaystyle
v^{(l)}=\sum_{i=1}^{l-1}a_i\,,\;\;$ ($v^{(0)}=0$ and $v^{(1)}=1$)
it is immediate to prove that if two paths $\mu$ and $\nu$ merge
at the level $l$ we have ${\rm Av}(E_\mu E_\mu) = v^{(l)}$. For
the Derrida-Gardner process over a tree ${\cal T}_{n,N}$ we will
use the symbol $\{{\cal E}, {\cal T}_{n,N}\}$.

Our theorem on the existence of the thermodynamical limit can be
used to show that the thermodynamical limit exist for the GREM in
the sense that if  $\{{\cal E}, {\cal T}_{n,N}\}$ is assigned for
a given $n$ and all $N>n$ we may show that its free energy density
is decreasing (and bounded) in $N$. In order to do so, starting
from a process $\{{\cal E}, {\cal T}_{n,N_1}\}$ we build the
process $\{{\cal E}^{(1)}_{\pi_1}, {\cal T}_{n,N}\}$ with
$N=N_1+N_2$ in the following way: at each vertex of the tree
${\cal T}_{n,N_1}$ sitting on the layer $i$ we increase the
multiplicity of the {\it furcation} by a factor $(\alpha_i)^{N_2}$
carrying {\rm the same value} $\epsilon_i^{(1)}$ to the newly
introduced branches. The new process will enjoy the property \be
{\rm Av}(E^{(1)}_{\pi_1(\sigma)}E^{(1)}_{\pi_1(\tau)}) \ge v^{(l)}
\; . \label{c1} \ee We apply the same construction to build
$\{{\cal E}^{(2)}_{\pi_2}, {\cal T}_{n,N}\}$ and we have \be {\rm
Av}(E^{(2)}_{\pi_2(\sigma)}E^{(2)}_{\pi_2(\tau)}) \ge v^{(l)} \; .
\label{c2} \ee It is now straightforward to verify that conditions
$(\ref{c1})$ and $(\ref{c2})$ imply $(\ref{punct})$. \vskip
12pt\noindent {\small {\bf Acknowledgments.}  One of us (P.C.)
thanks Francesco Guerra for useful conversations and Michael
Aizenman for introducing him to the Correlated Gaussian Random
Energy Models.
\par\noindent
This work has been partially supported by the European Commission
under the Research Training Network (Mathematical Aspects of
Quantum Chaos) n� HPRN-CT-2000-00103 of the IHP Programme.}

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