Simultaneous replica-symmetry breaking for vector spin glasses

TL;DR

This paper investigates the structure of mean-field vector spin glasses with non-convex interactions, showing that different spin types exhibit synchronized replica-symmetry-breaking behavior under certain conditions.

Contribution

It extends previous results by establishing a one-to-one correspondence of RSB structures across different spin types in vector spin glasses with non-convex interactions.

Findings

Replica-symmetry-breaking structures are synchronized across spin types.

Under modest assumptions, critical points of the functional describe the Gibbs measure.

Results generalize previous work on multi-species spherical spin glasses.

Abstract

We consider mean-field vector spin glasses with possibly non-convex interactions. Up to a small perturbation of the parameters defining the model, the asymptotic behavior of the Gibbs measure is described in terms of a critical point of an explicit functional. In this paper, we study some properties of these critical points. Under modest assumptions ensuring that different types of spins interact, we show that the replica-symmetry-breaking structures of the different types of spins are in one-to-one correspondence with one another. For instance, if some type of spins displays one level of replica-symmetry breaking, then so do all the other types of spins. This extends the recent results of [Electronic Journal of Probability, 27:1-75, 2022] and [Comm. Math. Phys., 394(3):1101-1152, 2022] that were obtained in the case of multi-species spherical spin glasses with convex interactions.