Spin glasses and Stein's method

TL;DR

This paper applies Stein's method to analyze high temperature spin glasses, providing new insights and limit theorems with error bounds, and offers a clearer understanding of the TAP equations.

Contribution

It introduces Stein's method for spin glasses, enabling direct Gibbs measure analysis and deriving limit theorems with total variation bounds, along with a novel explanation of TAP equations.

Findings

Stein's method can analyze Gibbs measures without cavity construction.

Limit theorems with total variation bounds are established.

A transparent derivation of TAP equations is provided.

Abstract

We introduce some applications of Stein's method in the high temperature analysis of spin glasses. Stein's method allows the direct analysis of the Gibbs measure without having to create a cavity. Another advantage is that it gives limit theorems with total variation error bounds, although the bounds can be suboptimal. A surprising byproduct of our analysis is a relatively transparent explanation of the Thouless-Anderson-Palmer system of equations. Along the way, we develop Stein's method for mixtures of two Gaussian densities.