Belief Propagation on the random $k$-SAT model

TL;DR

This paper proves that Belief Propagation accurately estimates the partition function in the random $k$-SAT model under certain conditions, revealing a phase transition related to replica symmetry breaking at low temperatures.

Contribution

It rigorously confirms Belief Propagation's effectiveness in the random $k$-SAT model and identifies a phase transition at low temperatures near the satisfiability threshold.

Findings

BP approximates the partition function well under replica symmetry conditions.

A phase transition occurs at low temperature near the satisfiability threshold.

The result aligns with predictions from statistical physics.

Abstract

Corroborating a prediction from statistical physics, we prove that the Belief Propagation message passing algorithm approximates the partition function of the random $k$-SAT model well for all clause/variable densities and all inverse temperatures for which a modest absence of long-range correlations condition is satisfied. This condition is known as "replica symmetry" in physics language. From this result we deduce that a replica symmetry breaking phase transition occurs in the random $k$-SAT model at low temperature for clause/variable densities below but close to the satisfiability threshold.