On zeros and algorithms for disordered systems: mean-field spin glasses

TL;DR

This paper develops deterministic quasipolynomial algorithms for accurately estimating partition functions in mean-field spin glasses, leveraging the analysis of zeros of the partition function across different models and phases.

Contribution

It introduces new algorithms that work efficiently across the entire replica-symmetric phase by studying the zeros of the partition function in mean-field spin glasses.

Findings

Algorithms succeed in the entire replica-symmetric phase.

Methods apply to both spherical and Ising spin models.

Achieve high-accuracy partition function estimation in polynomial time.

Abstract

Spin glasses are fundamental probability distributions at the core of statistical physics, the theory of average-case computational complexity, and modern high-dimensional statistical inference. In the mean-field setting, we design deterministic quasipolynomial-time algorithms for estimating the partition function to arbitrarily high accuracy for all inverse temperatures in the second moment regime. In particular, for the Sherrington--Kirkpatrick model, our algorithms succeed for the entire replica-symmetric phase. To achieve this, we study the locations of the zeros of the partition function. Notably, our methods are conceptually simple, and apply equally well to the spherical case and the case of Ising spins.