Some Rigorous Results on the L\'evy Spin Glass Model

Abstract

We study the L\'evy spin glass model, a fully connected model on $N$ vertices with heavy-tailed interactions governed by a power law distribution of order $0<\alpha<2.$ Our investigation is divided into three cases $0<\alpha<1$, $\alpha=1$, and $1<\alpha<2.$ When $1<\alpha<2,$ we identify a high temperature regime, in which the limit and fluctuation of the free energy are explicitly obtained and the site and bond overlaps are shown to exhibit concentration, interestingly, while the former is concentrated around zero, the latter obeys a positivity behavior. At any temperature, we further establish the existence of the limiting free energy and derive a variational formula analogous to Panchenko's framework in the setting of the Poissonian Viana-Bray model. For $\alpha=1$, the free energy scales super-linearly and converges to a constant proportional to $\beta$ in probability at any temperature. In the case of $0<\alpha<1$, the scaling for the free energy is again super-linear, however, it converges weakly to the sum of a Poisson Point Process at any temperature. Additionally, we show that the Gibbs measure puts most of its mass on the configurations that align with signs of the polynomially many heaviest edge weights.