Vector spin glasses with Mattis interaction II: non-convex high-temperature models

TL;DR

This paper proves that in high-temperature vector spin glass models with non-convex interactions, the free energy can be characterized by solutions to a Hamilton-Jacobi PDE, extending understanding beyond convex cases.

Contribution

It establishes the validity of a PDE-based description of free energy in non-convex high-temperature vector spin glasses, providing explicit formulas and large deviation principles.

Findings

Validates the PDE conjecture for non-convex models at high temperature

Provides explicit free energy representation via critical points

Derives large deviation principles for magnetization

Abstract

This paper constitutes the second 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 does not satisfy the usual convexity assumption. In this case, the Parisi formula breaks down, and there are no known methods to fully identify the limit free energy. It was suggested in [arXiv:1906.08471] that the limit free energy may be described using the unique solution of a partial differential equation of Hamilton--Jacobi type. In the present paper, we prove the validity of this conjecture in the high-temperature regime and provide an explicit representation for the free energy in terms of critical points. Using the duality between the free energy and large deviation principles, one can then easily deduce from the previous result a large deviation principle for the mean magnetization as well as a representation for the free energy of spin glass models with additional Mattis interaction at high temperature. In the companion paper, we establish similar results at all temperatures for models whose spin glass part is assumed to satisfy the usual convexity assumption.