Replica analysis of Franz-Parisi potential for sparse systems

TL;DR

This paper introduces a replica method to analyze the Franz-Parisi potential in sparse spin glass models, revealing connections to 1RSB cavity equations and aiding in transition temperature evaluation.

Contribution

It presents a novel replica symmetric approach to compute the Franz-Parisi potential for sparse systems, linking it to 1RSB cavity equations.

Findings

Derived self-consistent equations with multi-body overlaps

Equivalence to 1RSB cavity equations at x=1

Facilitates transition temperature estimation for p-spin models

Abstract

We propose a method for calculating the Franz-Parisi potential for spin glass models on sparse random graphs using the replica method under the replica symmetric ansatz. The resulting self-consistent equations have the solution with the characteristic structure of multi-body overlaps, and the self-consistent equations under this solution are equivalent to the one-step replica symmetry breaking (1RSB) cavity equation with Parisi parameter $x=1$. This method is useful for the evaluation of transition temperatures of the $p$-spin model on regular random graphs under a uniform magnetic field.