un 2 00 2 Thermodynamical Limit for Correlated Gaussian Random Energy Models

Let {Eσ(N)}σ∈ΣN be a family of |ΣN | = 2N centered unit Gaussian random variables defined by the covariance matrix CN of elements cN (σ, τ) := Av (Eσ(N)Eτ (N)), and HN (σ) = − √ NEσ(N) the corresponding random Hamiltonian. Then the quenched thermodynamical limit exists if, for every decomposition N = N1 + N2, and all pairs (σ, τ) ∈ ΣN × ΣN : cN (σ, τ) ≤ N1 N cN1(π1(σ), π1(τ)) + N2 N cN2(π2(σ), π2(τ)) where πk(σ), k = 1, 2 are the projections of σ ∈ ΣN into ΣNk . The condition is explicitly verified for the Sherrington-Kirckpatrick, the even p-spin, the Derrida REM and the Derrida-Gardner GREM models.