Free energy of bipartite spherical Sherrington--Kirkpatrick model

TL;DR

This paper analyzes the free energy of the bipartite spherical SK model, identifying the critical temperature, deriving the limiting free energy, and characterizing fluctuation distributions across temperature regimes.

Contribution

It determines the critical temperature, proves the limiting free energy for all non-critical temperatures, and establishes universal fluctuation laws including Gaussian and Tracy--Widom distributions.

Findings

Critical temperature identified

Limiting free energy derived for all non-critical temperatures

Fluctuations follow Gaussian above and Tracy--Widom below the critical temperature

Abstract

We consider the free energy of the bipartite spherical Sherrington--Kirkpatrick model. We find the critical temperature and prove the limiting free energy for all non-critical temperature. We also show that the law of the fluctuation of the free energy converges to the Gaussian distribution when the temperature is above the critical temperature, and to the GOE Tracy--Widom distribution when the temperature is below the critical temperature. The result is universal, and the analysis is applicable to a more general setting including the case where the disorders are non-identically distributed.