Short-range spin glasses and Random Overlap Structures

TL;DR

This paper investigates the properties of Random Overlap Structures in short-range spin glasses, demonstrating their local stability and establishing stochastic stability for the Edwards-Anderson model across all temperatures.

Contribution

It introduces a new approach to analyze the EA spin glass by leveraging ROSts, showing local stability exceeds stochastic stability, and extends existing theorems on pure state decomposition.

Findings

ROSts of the EA model have strong local stability.

Stochastic stability is proven for the EA model at all temperatures.

Extended theorems on pure state decomposition in spin glasses.

Abstract

Properties of Random Overlap Structures (ROSt)'s constructed from the Edwards-Anderson (EA) Spin Glass model on $\Z^d$ with periodic boundary conditions are studied. ROSt's are $\N\times\N$ random matrices whose entries are the overlaps of spin configurations sampled from the Gibbs measure. Since the ROSt construction is the same for mean-field models (like the Sherrington-Kirkpatrick model) as for short-range ones (like the EA model), the setup is a good common ground to study the effect of dimensionality on the properties of the Gibbs measure. In this spirit, it is shown, using translation invariance, that the ROSt of the EA model possesses a local stability that is stronger than stochastic stability, a property known to hold at almost all temperatures in many spin glass models with Gaussian couplings. This fact is used to prove stochastic stability for the EA spin glass at all temperatures and for a wide range of coupling distributions. On the way, a theorem of Newman and Stein about the pure state decomposition of the EA model is recovered and extended.