Universality of the fluctuations of the free energy in generalized Sherrington-Kirkpatrick models and the log likelihood ratio in spiked Wigner models

TL;DR

This paper proves that the fluctuations of free energy in generalized SK models and the log likelihood ratio in spiked Wigner models are Gaussian in the high temperature regime, showing universality across different disorder distributions.

Contribution

It establishes the Gaussian nature and universality of fluctuations in these models using a multigraph expansion approach.

Findings

Fluctuations are Gaussian in the high temperature regime.

Universality holds regardless of disorder distribution, depending only on a few parameters.

The multigraph expansion provides a unified analysis method.

Abstract

We consider the fluctuations of the free energy in generalized Sherrington-Kirkpatrick models and the log likelihood ratio of spiked Wigner models in the high temperature/subcritical regime. We prove that the limiting laws of the fluctuations are Gaussian under suitable assumptions, and the result is universal in the sense that it does not depend on the distribution of the disorder or the prior except that the means and the variances of the limiting laws depend on a few parameters of the model. The proof is based on the multigraph expansion that provides a unified approach to analyze both models.