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29 March 2024
 
  » arxiv » math-ph/0206007

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Thermodynamical Limit for Correlated Gaussian Random Energy Models
P. Contucci ; M. Degli Esposti ; C. Giardina ; S. Graffi ;
Date 6 Jun 2002
Subject Mathematical Physics; Disordered Systems and Neural Networks | math-ph cond-mat.dis-nn math.MP
AbstractLet ${E_{s}(N)}_{sinSigma_N}$ be a family of $|Sigma_N|=2^N$ centered unit Gaussian random variables defined by the covariance matrix $C_N$ of elements $displaystyle c_N(s, au):=av{E_{s}(N)E_{ au}(N)}$, 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 $(s, )in Sigma_N imes Sigma_N$: $$ c_N(s, au)leq frac{N_1}{N} c_{N_1}(pi_1(s),pi_1( au))+ frac{N_2}{N} c_{N_2}(pi_2(s),pi_2( au)) $$ where $pi_k(s), k=1,2$ are the projections of $sinSigma_N$ 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.
Source arXiv, math-ph/0206007
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