One-step replica symmetry breaking of random regular NAE-SAT I

TL;DR

This paper proves that in random regular k-NAE-SAT, solutions condense into a bounded number of clusters as predicted by the one-step replica symmetry breaking theory, confirming the physical conjecture about solution space structure.

Contribution

It provides a rigorous proof that solutions in random regular k-NAE-SAT condense into a finite number of clusters, validating the 1RSB prediction for this class of problems.

Findings

Most solutions lie within a bounded number of clusters.

Solution overlaps concentrate at two specific values.

Method based on moment analysis of an infinite spin system.

Abstract

In a broad class of sparse random constraint satisfaction problems(CSP), deep heuristics from statistical physics predict that there is a condensation phase transition before the satisfiability threshold, governed by one-step replica symmetry breaking(1RSB). In fact, in random regular k-NAE-SAT, which is one of such random CSPs, it was verified \cite{ssz22} that its free energy is well-defined and the explicit value follows the 1RSB prediction. However, for any model of sparse random CSP, it has been unknown whether the solution space indeed condenses on O(1) clusters according to the 1RSB prediction. In this paper, we give an affirmative answer to this question for the random regular k-NAE-SAT model. Namely, we prove that with probability bounded away from zero, most of the solutions lie inside a bounded number of solution clusters whose sizes are comparable to the scale of the free energy. Furthermore, we establish that the overlap between two independently drawn solutions concentrates precisely at two values. Our proof is based on a detailed moment analysis of a spin system, which has an infinite spin space that encodes the structure of solution clusters. We believe that our method is applicable to a broad range of random CSPs in the 1RSB universality class.