Multi-species mean-field spin-glasses. Rigorous results

TL;DR

This paper rigorously analyzes a multi-species spin glass model, establishing the thermodynamic limit, exploring various bounds, and examining replica symmetry breaking phenomena with a Parisi-like PDE.

Contribution

It extends spin glass theory to multi-species systems, proving the existence of the thermodynamic limit and analyzing replica symmetry breaking in this context.

Findings

Thermodynamic limit exists under convexity conditions.

Annealed approximation is exact at high temperatures.

Replica symmetry breaks down at low temperatures, showing negative entropy.

Abstract

We study a multi-species spin glass system where the density of each species is kept fixed at increasing volumes. The model reduces to the Sherrington-Kirkpatrick one for the single species case. The existence of the thermodynamic limit is proved for all densities values under a convexity condition on the interaction. The thermodynamic properties of the model are investigated and the annealed, the replica symmetric and the replica symmetry breaking bounds are proved using Guerra's scheme. The annealed approximation is proved to be exact under a high temperature condition. We show that the replica symmetric solution has negative entropy at low temperatures. We study the properties of a suitably defined replica symmetry breaking solution and we optimise it within a ziggurat ansatz. The generalized order parameter is described by a Parisi-like partial differential equation.