The convex structure of the Parisi formula for multi-species spin glasses

TL;DR

This paper demonstrates that the Parisi formula for multi-species spin glasses with convex covariance can be reformulated as a concave maximization problem, establishing the uniqueness of the solution and providing a new martingale-based representation.

Contribution

It introduces a transformation of the Parisi formula into a concave maximization over all probability measures, proving the uniqueness of the maximizer and offering a novel martingale-based free energy representation.

Findings

The Parisi formula can be expressed as a supremum over all probability measures of a concave functional.

The maximizer of the Parisi formula is unique.

A new representation of free energy as an infimum over martingales is derived.

Abstract

We study the free energy of mean-field multi-species spin glasses with convex covariance function. For such models with $D$ species, the Parisi formula is known to be valid, and expresses the limit free energy as a supremum over monotone probability measures on $\mathbb{R}_+^D$. We show here that one can transform this representation into a supremum over all probability measures on $\mathbb{R}_+^D$ of a concave functional. We then deduce that the Parisi formula admits a unique maximizer. Using convex-duality arguments, we also obtain a new representation of the free energy as an infimum over martingales in a Wiener space.