Order of fluctuations of the free energy in the positive semi-definite MSK model at critical temperature

TL;DR

This paper analyzes the fluctuations of free energy in the multi-species SK model at critical temperature, showing the variance scales as $O( ext{log}^2 N)$ when the variance profile is positive semi-definite.

Contribution

It extends previous results on the SK model to the multi-species case, establishing variance bounds at critical temperature with positive semi-definite variance profiles.

Findings

Variance of free energy is $O( ext{log}^2 N)$ at critical temperature.

Variance increases to $O( ext{log}^2 N + N^{1-eta})$ near the threshold.

Results generalize prior work on the SK model to multi-species models.

Abstract

In this note, we consider the multi-species Sherrington-Kirkpatrick spin glass model at its conjectured critical temperature, and we show that, when the variance profile matrix $\Delta^2$ is positive semi-definite, the variance of the free energy is $O(\log^2N)$. Furthermore, when one approaches this temperature threshold from the low temperature side at a rate of $O(N^{-\alpha})$ with $\alpha>0$, the variance is $O(\log^2N+N^{1-\alpha})$. This result is a direct extension of the work of Chen and Lam (2019) who proved an analogous result for the SK model, and our proof methods are adapted from theirs.