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.
Categories: math-ph cond-mat.dis-nn math.MP