Content-Type: multipart/mixed; boundary="-------------0306121050538" This is a multi-part message in MIME format. ---------------0306121050538 Content-Type: text/plain; name="03-277.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-277.keywords" Replica Simmetry Breaking, Thermodynamic Limit ---------------0306121050538 Content-Type: application/x-tex; name="mp-arc.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mp-arc.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}}} \def\ds{\displaystyle} %\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 CONVEX REPLICA SIMMETRY BREAKING\\ FROM POSITIVITY AND THERMODYNAMIC LIMIT} \vskip 1truecm \end{center} \begin{center}{Pierluigi Contucci, Sandro Graffi}\\ \vskip 1truecm {\small Dipartimento di Matematica} \\ {\small Universit\`a di Bologna, 40127 Bologna, Italy}\\ {\small {e-mail:contucci@dm.unibo.it, graffi@dm.unibo.it}} \end{center} % \vskip 1truecm %\begin{center} % \begin{abstract}\noindent Consider a correlated Gaussian random energy model built by successively adding one particle (spin) into the system and imposing the positivity of the associated covariance matrix. We show that the validity of a recently isolated condition ensuring the existence of the thermodynamic limit forces the covariance matrix to exhibit the Parisi replica symmetry breaking scheme with a convexity condition on the matrix elements. \end{abstract} \newpage The existence of the thermodynamic limit has been recently proved for the Sherrington-Kirckpatrick (SK) model \cite{1}, and more generally for any correlated Gaussian random energy (CGREM) model including the Derrida REM and the Derrida-Gardner GREM \cite{2}. In this letter we point out that the proof of \cite{2} may shed some light on the origin of Parisi's algebraic ansatz for the replica symmetry breaking (RSB) scheme which lies at the basis his solution \cite{3} of the SK model. Algebraically, Parisi's ansatz may be described as follows: start from the one by one matrix $q(0)>0$. Take $q(1)$ as $0c(0)$ To iterate the procedure, let us first describe the second step, i.e. the addition of a second spin to build a 2-spin system. As before, we first assume independence on the newly added spin variable. The covariance matrix turns then out to be (with the lexicographic order of the spin configurations) \be \tilde C(2)=\left( \begin{array}{cccc} {c(1)} & {c(1)} & {c(0)} & {c(0)} \cr {c(1)} & {c(1)} & {c(0)} & {c(0)} \cr {c(0)} & {c(0)} & {c(1)} & {c(1)} \cr {c(0)} & {c(0)} & {c(1)} & {c(1)} \cr \end{array}% \right) \; . \ee Again, this matrix is the covariance of a CGREM process only if it positive definite and non-degenerate. As above, this requires the dependence on the second spin variable. Among the possible ways to parametrize this dependence we choose the {\it minimal} one, namely the preceding one which only modifies the subprincipal diagonals: \be \bar C(2)=\left( \begin{array}{cccc} {c(1)} & {\bar q_2} & {c(0)} & {c(0)} \cr {\bar q_2} & {c(1)} & {c(0)} & {c(0)} \cr {c(0)} & {c(0)} & {c(1)} & {\bar q_2} \cr {c(0)} & {c(0)} & {\bar q_2} & {c(1)} \cr \end{array}% \right), \ee with $c(0)<\bar q_2c(1)>c(0)>0$. This last condition implies that the matrix is positive definite because the 4 principal minors are \be \Delta_1=c(2)>0\; , \ee \be \Delta_2=c(2)^2-c(1)^2>0 \; , \ee \be \Delta_3=[(c(2)-c(1)][c(2)(c(1)+c(2))-2c(0)^2]>0\; \ee and \be \Delta_4=[c(2) - c(1)]^2[((c(1)+c(2))^2 -4 c(0)^2]>0 \ee Corrispondigly with the spin representation we would have \be E_2(\s_1,\s_2) = \xi_0(2) + \xi_1(2)\s_1 + \xi_2(2)\s_2 + \xi_{1,2}\s_1\s_2 \; , \ee and \bea\nonumber Av(E_2(\s_1,\s_2)E_2(\t_1,\t_2)) &=& Av(\xi_0(2)^2) - Av(\xi_1(2)^2) - Av(\xi_2(2)^2) + Av(\xi_{1,2}(2)^2)\\ \nonumber &+& \delta_{\s_1,\t_1}[2Av(\xi_1(2)^2)-2Av(\xi_{1,2}(2)^2)] \\ \nonumber &+& \delta_{\s_2,\t_2}[2Av(\xi_2(2)^2)-2Av(\xi_{1,2}(2)^2)] \\ \nonumber &+& \delta_{\s_1,\t_1}\delta_{\s_2,\t_2} 4Av(\xi_{1,2}(2)^2) \; ; \eea and since we choose \be Av(\xi_2(2)^2)=2Av(\xi_{1,2}(2)^2) \ee the covariance matrix exibit the RSB scheme: \be Av(E_1(\s_1,\s_2)E_1(\t_1\t_2)) = a_0(2) + a_1(2)\delta_{\s_1,\s_2} + a_2(2)\delta_{\s_1,\t_1}\delta_{\s_2,\t_2} \; . \ee In general we will assume that our construction is done adding at each step the $N$-th spin variable and the newly added interaction terms are indipendent realization of the same Gaussian distribution or, in other terms, the distribution of the Gaussian random variables depends only on the last index. The general scheme is the described by a correlated Gaussian process \be E_N(\s) = \xi_0(N) + \sum_{1\le i\le N}\xi_i(N)\s_i + \sum_{1\le i