A Remark on the Infinite-Volume Gibbs Measures of Spin Glasses

TL;DR

This paper demonstrates that infinite-volume Gibbs measures of spin glasses can be represented as random probability measures on a Hilbert space, providing a new perspective for analyzing their structure and properties.

Contribution

It introduces a novel approach to study Gibbs measures as measures on a Hilbert space, extending the space of Random Overlap Structures and connecting with existing frameworks.

Findings

Infinite-volume Gibbs measures can be identified as measures on a Hilbert space.

If finite-volume Gibbs measures satisfy Ghirlanda-Guerra identities, the infinite-volume measure is singular.

The approach links Gibbs measures with the theory of random matrices and exchangeability.

Abstract

In this note, we point out that infinite-volume Gibbs measures of spin glass models on the hypercube can be identified as random probability measures on the unit ball of a Hilbert space. This simple observation follows from a result of Dovbysh and Sudakov on weakly exchangeable random matrices. Limiting Gibbs measures can then be studied as single well-defined objects. This approach naturally extends the space of Random Overlap Structures as defined by Aizenman, Sims and Starr. We discuss the Ruelle Probability Cascades and the stochastic stability within this framework. As an application, we use an idea of Parisi and Talagrand to prove that if a sequence of finite-volume Gibbs measures satisfies the Ghirlanda-Guerra identities, then the infinite-volume measure must be singular as a measure on a Hilbert space.