The cavity method for large deviations

TL;DR

This paper introduces an extension of the cavity method to analyze large deviations in disordered systems, enabling the computation of rare event probabilities in complex graph-based models with replica symmetry breaking.

Contribution

It develops a novel cavity method framework for large deviations, applicable to combinatorial optimization problems and adaptive graph models, accounting for replica symmetry breaking phases.

Findings

Successfully computes rate functions for vertex-cover and coloring problems.

Demonstrates the method's applicability to models on adaptive graph structures.

Provides insights into rare events in disordered systems with complex phase behavior.

Abstract

A method is introduced for studying large deviations in the context of statistical physics of disordered systems. The approach, based on an extension of the cavity method to atypical realizations of the quenched disorder, allows us to compute exponentially small probabilities (rate functions) over different classes of random graphs. It is illustrated with two combinatorial optimization problems, the vertex-cover and coloring problems, for which the presence of replica symmetry breaking phases is taken into account. Applications include the analysis of models on adaptive graph structures.