Fluctuations of the Free Energy of the Spherical Sherrington–Kirkpatrick Model with Heavy-Tailed Interaction

Abstract

We consider the 2-spin spherical Sherrington–Kirkpatrick model without external magnetic field where the interactions between the spins are given as random variables with heavy-tailed distribution. We show that the free energy exhibits a sharp phase transition depending on the location of the largest eigenvalue of the interaction matrix. We also prove the order of the limiting free energy and the limiting distribution of the fluctuation of the free energy for both regimes.