Replica symmetry breaking in the `small world' spin glass

TL;DR

This paper applies the cavity method to analyze a small world spin glass model, demonstrating improved accuracy over replica symmetric theory by incorporating one step replica symmetry breaking, and validating results with numerical simulations.

Contribution

It introduces a cavity method approach with 1RSB for small world spin glasses, extending previous Bethe lattice techniques to this model.

Findings

Results align better with simulations than RS theory

Demonstrates the effectiveness of 1RSB in small world spin glasses

Provides a new analytical framework for such models

Abstract

We apply the cavity method to a spin glass model on a `small world' lattice, a random bond graph super-imposed upon a 1-dimensional ferromagnetic ring. We show the correspondence with a replicated transfer matrix approach, up to the level of one step replica symmetry breaking (1RSB). Using the scheme developed by M\'ezard & Parisi for the Bethe lattice, we evaluate observables for a model with fixed connectivity and $\pm J$ long range bonds. Our results agree with numerical simulations significantly better than the replica symmetric (RS) theory.