Marginals of a spherical spin glass model with correlated disorder

TL;DR

This paper proves the convergence of finite marginals of a spherical spin glass model with correlated disorder in high-temperature regimes, providing explicit asymptotic descriptions and convergence rates.

Contribution

It introduces a rigorous analysis of the marginals' convergence for correlated spherical spin glasses, including explicit asymptotic measures and convergence rate bounds.

Findings

Finite marginals converge to an explicit decoupled measure

Established upper bounds for convergence rates

Derived a concentration inequality for bounded functions

Abstract

In this paper we prove the weak convergence, in a high-temperature phase, of the finite marginals of the Gibbs measure associated to a symmetric spherical spin glass model with correlated couplings towards an explicit asymptotic decoupled measure. We also provide upper bounds for the rate of convergence in terms of the one of the energy per variable. Furthermore, we establish a concentration inequality for bounded functions under a higher temperature condition. These results are exemplified by analysing the asymptotic behaviour of the empirical mean of coordinate-wise functions of samples from the Gibbs measure of the model.