Fluctuations of the free energy of the mixed $p$-spin mean field spin glass model

TL;DR

This paper proves that at high temperatures, the free energy fluctuations in the mixed p-spin spin glass model follow a Gaussian distribution, driven by cycle counts weighted by the 2-spin interactions.

Contribution

It establishes the Gaussian fluctuation limit for the free energy of the mixed p-spin model with a non-zero 2-spin component at high temperature.

Findings

Fluctuations converge in distribution to a Gaussian.

Fluctuations originate from weighted cycle counts in the interaction graph.

Results apply to models with non-vanishing 2-spin component at high temperature.

Abstract

We prove the convergence in distribution of the fluctuations of the free energy of the mixed $p$-spin Sherrington-Kirkpatrick model with non-vanishing $2$-spin component at high enough temperature. The limit is Gaussian, and the fluctuations are seen to arise from weighted cycle counts in the complete graph on the spin indices weighted by the $2$-spin interaction matrix.