Optimisation problems and replica symmetry breaking in finite connectivity spin-glasses

TL;DR

This paper introduces a formalism for handling the first step of hierarchical replica symmetry breaking in finite-connectivity spin-glasses, applying it to optimize the understanding of the random 3-Satisfiability problem.

Contribution

It develops a new formalism for the first step of replica symmetry breaking in finite-connectivity models, extending previous methods and applying it to optimization problems.

Findings

First RSB solution for 3-Satisfiability improves upon replica symmetric results.

Formalism coincides with previous models in symmetric and limiting cases.

Provides a probability distribution-based order parameter for hierarchical RSB.

Abstract

A formalism capable of handling the first step of hierarchical replica symmetry breaking in finite-connectivity models is introduced. The emerging order parameter is claimed to be a probability distribution over the space of field distributions (or, equivalently magnetisation distributions) inside the cluster of states. The approach is shown to coincide with the previous works in the replica symmetric case and in the two limit cases m=0,1 where m is Parisi's break-point. As an application to the study of optimization problems, the ground-state properties of the random 3-Satisfiability problem are investigated and we present a first RSB solution improving replica symmetric results.