Clusters of solutions and replica symmetry breaking in random k-satisfiability

TL;DR

This paper analyzes the structure of solutions in random k-satisfiability problems using the cavity method, identifying phase transitions and solution clustering behavior, with new theoretical insights and refined phase transition locations.

Contribution

It provides a refined analysis of the clustering and condensation transitions in random k-satisfiability, including precise transition locations and a simplified cavity formalism for special parameters.

Findings

Determined the exact location of the clustering transition.

Uncovered a second condensation phase transition for k ≥ 4.

Developed a simplified cavity formalism and new large-k expansions.

Abstract

We study the set of solutions of random k-satisfiability formulae through the cavity method. It is known that, for an interval of the clause-to-variables ratio, this decomposes into an exponential number of pure states (clusters). We refine substantially this picture by: (i) determining the precise location of the clustering transition; (ii) uncovering a second `condensation' phase transition in the structure of the solution set for k larger or equal than 4. These results both follow from computing the large deviation rate of the internal entropy of pure states. From a technical point of view our main contributions are a simplified version of the cavity formalism for special values of the Parisi replica symmetry breaking parameter m (in particular for m=1 via a correspondence with the tree reconstruction problem) and new large-k expansions.