Thermodynamical limit for correlated Gaussian random energy models

Let{E-sigma(N)}(sigmais an element ofSigmaN) be a family Of \E-N\ = 2(N) centered unit Gaussian random variables defined by the covariance matrix C-N of elements C-N(sigma, tau):=Av(E-sigma(N)E-tau(N)) and H-N (sigma) = -rootNE(sigma) (N) the corresponding random Hamiltonian. Then the quenched thermodynamical limit exists if, for every decomposition N = N-1 + N-2, and all pairs (sigma, tau) is an element of Sigma(N) X Sigma(N): C-N(sigma, tau) less than or equal to N-1/N c(N1) (pi(1)(sigma), pi(1)(sigma), pi(1)(tau)) + - N-2/N c(N2) (pi(2) (pi(2)(sigma), pi(2) (tau)), where pi(k) (sigma), k = 1, 2 are the projections of or sigma is an element of Sigma(N) into Sigma(Nk). The condition is explicitly verified for the Sherrington-Kirkpatrick, the even p-spin, the Derrida REM and the Derrida-Gardner GREM models.