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

TL;DR

The paper addresses the order of fluctuations of the free energy in the multi-species SK model at the conjectured critical temperature. It extends Chen and Lam's variance-bounding approach from the single-species SK model to the MSK setting by leveraging Chatterjee's variance identity and a two-parameter interpolation, under the assumption that the variance-profile matrix $\Delta^2$ is positive semi-definite. The main results are that $\mathrm{Var}(F_N(\beta_c)) = O((\log N)^2)$ and, near criticality with $\beta^2 = \beta_c^2 + d N^{-\alpha}$, $\mathrm{Var}(F_N) = O((\log N)^2 + N^{1-\alpha})$, with the rank $r$ of $\Delta^2$ entering the bound via a $x$-dependent control on the exponential moments of the multi-overlap. The approach highlights the role of the multi-overlaps and PSD structure in controlling fluctuations, providing a rigorous pathway to generalizing SK results to MSK and informing expectations for Parisi-type descriptions in the PSD MSK regime.

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 $Δ^2$ 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 $O(N^{-α})$ with $α>0$, the variance is $O(\log^2N+N^{1-α})$. 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.

Theorems & Definitions (7)

• Definition 1.1: MSK model
• Theorem 1.2
• Lemma 2.1
• proof : Proof of Theorem \ref{['thm:maintheorem']}
• proof : Proof of Lemma \ref{['lem:mainlemma']}
• Lemma 2.2
• proof : Proof of Lemma \ref{['lem:Talagrand_bound']}