The condensation phase transition in the regular $k$-SAT model

TL;DR

This paper rigorously proves the existence and location of a condensation phase transition in the random regular k-SAT model by directly leveraging Belief Propagation, bridging physics predictions and mathematical validation.

Contribution

It introduces a novel method to use Belief Propagation for rigorous proofs of phase transitions in random CSPs, specifically in the regular k-SAT model.

Findings

Confirmed the condensation phase transition in the regular k-SAT model.

Determined the precise location of the phase transition.

Bridged physics predictions with rigorous mathematical proof.

Abstract

Much of the recent work on random constraint satisfaction problems has been inspired by ingenious but non-rigorous approaches from physics. The physics predictions typically come in the form of distributional fixed point problems that are intended to mimic Belief Propagation, a message passing algorithm, applied to the random CSP. In this paper we propose a novel method for harnessing Belief Propagation directly to obtain a rigorous proof of such a prediction, namely the existence and location of a condensation phase transition in the random regular $k$-SAT model.