Vector spin glasses with Mattis interaction I: the convex case

TL;DR

This paper analyzes convex vector spin glass models with Mattis interactions, deriving a Parisi formula for the limit free energy and establishing a large deviation principle for mean magnetization, simplifying previous methods.

Contribution

It provides a new, simplified proof for the limit free energy and large deviations in convex vector spin glasses with Mattis interactions, treating the interaction as a parameter.

Findings

Identified the limit free energy via a Parisi-type formula.

Proved a large deviation principle for the mean magnetization.

Simplified the proof approach compared to previous methods.

Abstract

This paper constitutes the first part of a two-paper series devoted to the systematic study of vector spin glass models whose energy function involves a spin glass part and a general Mattis interaction part. In this paper, we focus on models whose spin glass part satisfies the usual convexity assumption. We identify the limit free energy via a Parisi-type formula and prove a large deviation principle for the mean magnetization. The proof is remarkably simple and short compared to previous approaches; it relies on treating the Mattis interaction as a parameter of the model. In the companion paper, we establish similar results in the high-temperature regime for models whose spin glass part is not assumed to satisfy the usual convexity assumption.