Thermodynamical Limit for Correlated Gaussian Random Energy Models

 Let {E _Σ (N)} _ΣΣN be a family of |Σ _N |=2 ^N centered unit Gaussian random variables defined by the covariance matrix C _N of elements c _N (Σ,τ):=Av(E _Σ (N)E _τ (N)) and the corresponding random Hamiltonian. Then the quenched thermodynamical limit exists if, for every decomposition N=N ₁ +N ₂ , and all pairs (Σ,τ)Σ _N ×Σ _N : where π _k (Σ),k=1,2 are the projections of ΣΣ _N into Σ _Nk . The condition is explicitly verified for the Sherrington-Kirkpatrick, the even p-spin, the Derrida REM and the Derrida-Gardner GREM models.