On the phase diagram of the multiscale mean-field spin-glass

TL;DR

This paper explores the phase diagram of a multiscale SK spin-glass model with couplings at different time scales, providing rigorous results on overlap moments, replica symmetry, and conditions for gaps in order parameters.

Contribution

It introduces a multiscale generalization of the SK model, computes the second moment of overlaps, and establishes conditions for replica symmetry breaking and support gaps.

Findings

Computed the asymptotic second moment of overlaps.

Identified conditions for replica symmetry and symmetry breaking.

Proved the existence of gaps in the order parameter support.

Abstract

In this paper we study the phase diagram of a Sherrington-Kirkpatrick (SK) model where the couplings are forced to thermalize at different time scales. Besides being a challenging generalization of the SK model, such settings may arise naturally in physics whenever part of the many degrees of freedom of a system relaxes to equilibrium considerably faster than the others. For this model we compute the asymptotic value of the second moment of the overlap distribution. Furthermore, we provide a rigorous sufficient condition for an annealed solution to hold, identifying a high temperature, or weak coupling, region. In addition, we also prove that for sufficiently strong couplings the solution must present a number of replica symmetry breaking levels at least equal to the number of time scales already present in the multiscale model. Finally, we give a sufficient condition for the existence of gaps in the support of the functional order parameters.