Metastable configurations on the Bethe lattice

TL;DR

This paper introduces an analytic cavity method to calculate the number of metastable states in Ising spin systems on the Bethe lattice, covering ferromagnetic and spin glass interactions, with insights into replica symmetry breaking.

Contribution

It develops a general analytic approach using the cavity method to analyze metastable configurations in Bethe lattice spin systems, including spin glasses.

Findings

Computed metastable state counts for ferromagnetic and spin glass models

Demonstrated the application of replica symmetry breaking techniques

Provided analytical insights into energy landscapes of Bethe lattice systems

Abstract

We present a general analytic method to compute the number of metastable configurations as a function of the energy for a system of interacting Ising spins on the Bethe lattice. Our approach is based on the cavity method. We apply it to the case of ferromagnetic interactions, and also to the binary and Gaussian spin glasses. Most of our results are obtained within the replica symmetric ansatz, but we illustrate how replica symmetry breaking can be performed.