Thermodynamical Limit for Correlated Gaussian Random Energy Models

Abstract

Let $\{E_{\s}(N)\}_{\s\inΣ_N}$ be a family of $|Σ_N|=2^N$ centered unit Gaussian random variables defined by the covariance matrix $C_N$ of elements $\displaystyle c_N(\s,τ):=\av{E_{\s}(N)E_τ(N)}$, 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 $(\s,\t)\in Σ_N\times Σ_N$: $$ c_N(\s,τ)\leq \frac{N_1}{N} c_{N_1}(π_1(\s),π_1(τ))+ \frac{N_2}{N} c_{N_2}(π_2(\s),π_2(τ)) $$ where $π_k(\s), k=1,2$ are the projections of $\s\inΣ_N$ into $Σ_{N_k}$. The condition is explicitly verified for the Sherrington-Kirckpatrick, the even $p$-spin, the Derrida REM and the Derrida-Gardner GREM models.