Properties of the full replica symmetry breaking free energy functional of the Ising spin glass on random regular graph

TL;DR

This paper analyzes the mathematical properties of the full replica symmetry breaking free energy functional for the Ising spin glass on random regular graphs, focusing on auxiliary variational problems and their solutions.

Contribution

It provides a detailed study of the auxiliary variational problems, including existence, uniqueness, and connection to discrete replica symmetry breaking solutions.

Findings

Derived the self-consistency equation for the auxiliary order parameter.

Proved existence and uniqueness of solutions to the variational problems.

Showed the functional recovers discrete replica symmetry breaking solutions under certain conditions.

Abstract

The full replica symmetry breaking free energy of the Ising spin glass on random regular graphs is given by the solutions of two auxiliary variational problems inside a global (physical) variational problem on the order parameter. In this paper, we provide a detailed study of the auxiliary variational problems. We get the self-consistency equation for the auxiliary order parameter and obtain the existence and uniqueness results. We also show that the full replica symmetry breaking free energy functional recovers the discrete replica symmetry breaking solutions, by imposing some proper conditions on the physical order parameter.