P. Contucci, M. Degli Esposti, C. Giardina', S. Graffi
Thermodynamical Limit for Correlated Gaussian Random Energy Models
(25K, LaTeX 2e)

ABSTRACT.  Let $\{E_{\s}(N)\}_{\s\in\Sigma_N}$ be a family of $|\Sigma_N|=2^N$ 
centered unit Gaussian random variables defined by the covariance 
matrix $C_N$ of elements $\displaystyle 
 c_N(\s,\tau):=\av{E_{\s}(N)E_{\tau}(N)}$, 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 $(\s,\t)\in \Sigma_N\times \Sigma_N$: 
$$ 
c_N(\s,\tau)\leq \frac{N_1}{N}\;c_{N_1}(\pi_1(\s),\pi_1(\tau))+ 
\frac{N_2}{N}\;c_{N_2}(\pi_2(\s),\pi_2(\tau)) 
$$ 
where $\pi_k(\s), k=1,2$ are the projections of $\s\in\Sigma_N$ 
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.