On one-step replica symmetry breaking in the Edwards-Anderson spin glass model

TL;DR

This paper develops a one-step replica symmetry breaking framework for the 2D Edwards-Anderson spin glass model using Kikuchi approximations and Generalized Belief Propagation, revealing new equations and solution classes.

Contribution

It introduces a novel 1RSB GBP approach for the 2D EA model, extending replica symmetry breaking methods beyond tree-like graphs.

Findings

Derived a generalized free energy incorporating Kikuchi approximation and Parisi parameter y.

Formulated 1RSB GBP equations that generalize belief propagation for the EA model.

Identified conditions for simpler solution classes analogous to Survey Propagation.

Abstract

We consider a one-step replica symmetry breaking description of the Edwards-Anderson spin glass model in 2D. The ingredients of this description are a Kikuchi approximation to the free energy and a second-level statistical model built on the extremal points of the Kikuchi approximation, which are also fixed points of a Generalized Belief Propagation (GBP) scheme. We show that a generalized free energy can be constructed where these extremal points are exponentially weighted by their Kikuchi free energy and a Parisi parameter $y$, and that the Kikuchi approximation of this generalized free energy leads to second-level, one-step replica symmetry breaking (1RSB), GBP equations. We then proceed analogously to Bethe approximation case for tree-like graphs, where it has been shown that 1RSB Belief Propagation equations admit a Survey Propagation solution. We discuss when and how the one-step-replica symmetry breaking GBP equations that we obtain also allow a simpler class of solutions which can be interpreted as a class of Generalized Survey Propagation equations for the single instance graph case.