Some Rigorous Results on the Lévy Spin Glass Model

Abstract

We study the Lévy spin glass model, a fully connected model on N vertices with heavy-tailed interactions governed by a power law distribution of order 0 < α < 2. Our investigation focuses on two domains 0 < α < 1 and 1 < α < 2. When 1 < α < 2, we identify a high temperature regime, in which the limit and fluctuation of the free energy are explicitly obtained and the site and bound 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. In the case of 0 < α < 1, we show that the Lévy model behaves differently, where the proper scaling for the free energy is N 1/α instead of N and the normalized free energy converges weakly to the sum of a Poisson Point Process at any temperature. Additionally, we show that the Gibbs measure carries its weight on polynomially many heaviest edges.