Some Properties of the Phase Diagram for Mixed $p$-Spin Glasses

TL;DR

This paper analyzes the phase diagram of mixed p-spin glasses using the Parisi variational framework, deriving conditions for replica symmetry breaking and conjecturing the phase boundary in the temperature-external field plane.

Contribution

It introduces a characterization of Parisi measures via first order optimality, generalizes de Almeida-Thouless conditions, and conjectures the phase boundary for all models.

Findings

Replica symmetric phase characterized by de Almeida-Thouless condition.

Complement of RS region is in the RSB phase.

Phase boundary matches the conjecture except for a bounded set.

Abstract

In this paper we study the Parisi variational problem for mixed $p$-spin glasses with Ising spins. Our starting point is a characterization of Parisi measures whose origin lies in the first order optimality conditions for the Parisi functional, which is known to be strictly convex. Using this characterization, we study the phase diagram in the temperature-external field plane. We begin by deriving self-consistency conditions for Parisi measures that generalize those of de Almeida and Thouless to all levels of Replica Symmetry Breaking (RSB) and all models. As a consequence, we conjecture that for all models the Replica Symmetric (RS) phase is the region determined by the natural analogue of the de Almeida-Thouless condition. We show that for all models, the complement of this region is in the RSB phase. Furthermore, we show that the conjectured phase boundary is exactly the phase boundary in the plane less a bounded set. In the case of the Sherrington-Kirkpatrick model, we extend this last result to show that this bounded set does not contain the critical point at zero external field.