The condensation transition in random hypergraph 2-coloring

TL;DR

This paper proves the existence of a condensation phase transition in random hypergraph 2-coloring, revealing the limitations of the second moment method and providing improved bounds on the problem's threshold.

Contribution

It is the first rigorous proof of a condensation transition in a natural random CSP, specifically in hypergraph 2-coloring, and it shows the second moment method fails before this transition.

Findings

Condensation transition exists in random hypergraph 2-coloring.

Second moment method breaks down before the condensation transition.

Improved bounds on the hypergraph 2-colorability threshold.

Abstract

For many random constraint satisfaction problems such as random satisfiability or random graph or hypergraph coloring, the best current estimates of the threshold for the existence of solutions are based on the first and the second moment method. However, in most cases these techniques do not yield matching upper and lower bounds. Sophisticated but non-rigorous arguments from statistical mechanics have ascribed this discrepancy to the existence of a phase transition called condensation that occurs shortly before the actual threshold for the existence of solutions and that affects the combinatorial nature of the problem (Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborova: PNAS 2007). In this paper we prove for the first time that a condensation transition exists in a natural random CSP, namely in random hypergraph 2-coloring. Perhaps surprisingly, we find that the second moment method breaks down strictly \emph{before} the condensation transition. Our proof also yields slightly improved bounds on the threshold for random hypergraph 2-colorability. We expect that our techniques can be extended to other, related problems such as random k-SAT or random graph k-coloring.