Free energy of the bipartite spherical SK model at critical temperature

TL;DR

This paper analyzes the critical temperature behavior of the bipartite spherical SK model, revealing a transition in free energy fluctuations characterized by a sum of Gaussian and Tracy-Widom variables within a specific temperature window.

Contribution

It extends the understanding of free energy fluctuations at critical temperature to the bipartite SSK model, identifying a transitional fluctuation regime similar to the single-party case.

Findings

Free energy fluctuations are Gaussian or Tracy-Widom at high/low temperatures.

A transitional window of size n^{-1/3}√log n around critical temperature is identified.

Within this window, fluctuations are a sum of Gaussian and Tracy-Widom variables.

Abstract

The spherical Sherrington-Kirkpatrick (SSK) model and its bipartite analog both exhibit the phenomenon that their free energy fluctuations are asymptotically Gaussian at high temperature but asymptotically Tracy-Widom at low temperature. This was proved in two papers by Baik and Lee, for all non-critical temperatures. The case of critical temperature was recently computed for the SSK model in two separate papers, one by Landon and the other by Johnstone, Klochkov, Onatski, Pavlyshyn. In the current paper, we derive the critical temperature result for the bipartite SSK model. In particular, we find that the free energy fluctuations exhibit a transition when the temperature is in a window of size $n^{-1/3}\sqrt{\log n}$ around the critical temperature, the same window for the SSK model. Within this transitional window, the asymptotic fluctuations of the free energy are the sum of independent Gaussian and Tracy-Widom random variables.