Some Observations for Mean-Field Spin Glass Models

TL;DR

This paper establishes bounds on the pressure of mean-field spin glass models, demonstrating Lipschitz continuity with respect to the distribution of couplings, and extends universality results to more general models.

Contribution

It provides new bounds for the pressure's dependence on coupling distributions and generalizes universality results to Viana-Bray models, including non-independent couplings.

Findings

Pressure is Lipschitz continuous in coupling distribution

Re-derivation of universality of SK model

Extension to Viana-Bray model with dependent couplings

Abstract

We obtain bounds to show that the pressure of a two-body, mean-field spin glass is a Lipschitz function of the underlying distribution of the random coupling constants, with respect to a particular semi-norm. This allows us to re-derive a result of Carmona and Hu, on the universality of the SK model, by a different proof, and to generalize this result to the Viana-Bray model. We also prove another bound, suitable when the coupling constants are not independent, which is what is necessary if one wants to consider ``canonical'' instead of ``grand canonical'' versions of the SK and Viana-Bray models. Finally, we review Viana-Bray type models, using the language of L\'evy processes, which is natural in this context.