Content-Type: multipart/mixed; boundary="-------------0206061112497" This is a multi-part message in MIME format. ---------------0206061112497 Content-Type: text/plain; name="02-256.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-256.keywords" GREM models, Thermodinamical limit ---------------0206061112497 Content-Type: application/x-tex; name="crem.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="crem.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[12pt,oneside]{article} \usepackage{amsfonts,amssymb,graphicx} \linespread{1.5} %% % ABBREVIAZIONI % \def\no{\noindent} \def\be{\begin{equation}} \def\ee{\end{equation}} \def\bea{\begin{eqnarray}} \def\eea{\end{eqnarray}} \def\<{\langle} \def\>{\rangle} \def\~{\tilde} \def\s{\sigma} \def\l{\lambda} \def\a{\alpha} \def\b{\beta} \def\g{\gamma} \def\o{\omega} \def\t{\tau} \def\n{\eta} \def\bs{{\bar{s}}} \def\hs{{\hat{s}}} %\newtheorem{theorem}{Theorem} %\newtheorem{proposition}[theorem]{Proposition} %\newtheorem{lemma}[theorem]{Lemma} %\newtheorem{corollary}[theorem]{Corollary} %\newtheorem{definition}[theorem]{Definition} %\newtheorem{remark}[theorem]{Remark} % %%%%% definizioni Mirko \newcommand{\brac}[1]{\< #1\>} \newcommand{\av}[1]{\mbox{{\rm Av}}\left(#1\right)} \newcommand{\dete}[1]{\mbox{det}\left(#1\right)} \newcommand{\zn}{Z_N} \newcommand{\nbs}{\sqrt{N}\beta E_\sigma} \newcommand{\nbt}{\sqrt{N}\beta E_\tau} \newcommand{\R}{\Bbb R} \newcommand{\C}{\Bbb C} \newcommand{\N}{\Bbb N} \newcommand{\Q}{\Bbb Q} \newcommand{\T}{\Bbb T} \newcommand{\Z}{\Bbb Z} \newcommand{\1}{\Bbb 1} \newtheorem{remark}{Remark} \newtheorem{proposition}{Proposition} \newtheorem{theorem}{THEOREM} \newtheorem{definition}{Definition} \newtheorem{corollary}{Corollary} \newenvironment{proof}{Proof:}{\hfill$\square$\vskip.5cm} %%%% fine definizioni Mirko % % \begin{document} % \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} % %\begin{center} % \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}\}\,\,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$ 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 $K_t$ 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}_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, $nn$ 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.} \begin{thebibliography}{GGGG} {\small \bibitem[BS]{BS} E.Bolthausen and A.S.Sznitman, ``On Ruelle's probability cascades and an abstract cavity method'' Commun. Math. 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