Parisi PDE and convexity for vector spins

TL;DR

This paper extends the Parisi formula and PDE framework to vector spin glasses with self-overlap correction, providing a new decomposition of the order parameter and establishing convexity properties of the Parisi functional.

Contribution

It introduces a novel decomposition of the Parisi path into a Lipschitz matrix path and a quantile function, and generalizes the Parisi PDE for vector spins.

Findings

Reformulation of the Parisi formula using a PDE for vector spins

Decomposition of the order parameter into a Lipschitz path and a quantile function

Proof of strict convexity of the Parisi functional for fixed Lipschitz paths

Abstract

We consider mean-field vector spin glasses with self-overlap correction. The limit of free energy is known to be the Parisi formula, which is an infimum over matrix-valued paths. We decompose such a path into a Lipschitz matrix-valued path and the quantile function of a one-dimensional probability measure. For such a pair, we associate a Parisi PDE generalized for vector spins. Under mild conditions, we rewrite the Parisi formula in terms of solutions of the PDE. Moreover, for each fixed Lipschitz path, the Parisi functional is strictly convex over probability measures.