Local independence in mean-field spin glasses

TL;DR

This paper introduces a new approach to understanding local independence in spin glasses, demonstrating that under certain conditions, subsets of spins become asymptotically independent, with implications for classical and soft spin models.

Contribution

It generalizes the cavity method by considering multiple cavity sites, establishing a link between local independence and replica-symmetric properties in spin glasses.

Findings

Asymptotic independence of fixed subsets of spins under replica-symmetric conditions

Equivalence between local independence and replica-symmetric properties

Framework applies to classical, soft spin, and Gardner spin glasses

Abstract

We present a new approach to local independence in spin glasses, i.e. the phenomenon that any fixed subset of coordinates is asymptotically independent in the thermodynamic limit. The approach generalizes the rigorous cavity method from Talagrand by considering multiple cavity sites. Under replica-symmetric conditions of thin-shell and overlap concentration, the cavity fields are revealed to be asymptotically independent, conditionally on the disorder, which in turn leads to local independence. Conversely, it is shown that local independence implies those replica-symmetric properties. The framework is general enough to encompass the classical and soft spin ($[-1,1]$) Sherrington-Kirkpatrick models, as well as the Gardner spin glasses.