Universality of Superconcentration in the Sherrington-Kirkpatrick Model

TL;DR

This paper demonstrates that superconcentration of the free energy variance in the SK model is universal across disorders with finite third moments, extending previous Gaussian-specific results.

Contribution

It generalizes superconcentration results from Gaussian disorders to a broader class with finite third moments, providing explicit bounds.

Findings

Superconcentration holds for any centered disorder with finite third moment.

Variance bounds are expressed in terms of an auxiliary function related to the disorder.

Under regularity conditions, variance is at most of order N/log N.

Abstract

We study the universality of superconcentration for the free energy in the Sherrington-Kirkpatrick (SK) model. In arXiv:0907.3381, Chatterjee showed that when the system consists of $N$ spins and Gaussian disorders, the variance of this quantity is superconcentrated by establishing an upper bound of order $N/\log{N}$, in contrast to the $O(N)$ bound obtained from the Gaussian-Poincar\'e inequality. In this paper, we show that superconcentration indeed holds for any choice of centered disorders with finite third moment, where the upper bound is expressed in terms of an auxiliary nondecreasing function $f$ that arises in the representation of the disorder as $f(g)$ for $g$ standard normal. Under an additional regularity assumption on $f$, we further show that the variance is of order at most $N/\log{N}$.