A positive temperature phase transition in random hypergraph 2-coloring

TL;DR

This paper rigorously proves the existence and asymptotic position of a phase transition in the 2-coloring problem of random hypergraphs, confirming physicists' predictions from the cavity method.

Contribution

It establishes the rigorous existence and precise asymptotic location of the condensation phase transition in random hypergraph 2-coloring.

Findings

Confirmed the phase transition predicted by the cavity method.

Determined the asymptotic location of the phase transition.

Provided a rigorous mathematical proof for the phenomenon.

Abstract

Diluted mean-field models are graphical models in which the geometry of interactions is determined by a sparse random graph or hypergraph. Based on a nonrigorous but analytic approach called the "cavity method", physicists have predicted that in many diluted mean-field models a phase transition occurs as the inverse temperature grows from $0$ to $\infty$ [Proc. National Academy of Sciences 104 (2007) 10318-10323]. In this paper, we establish the existence and asymptotic location of this so-called condensation phase transition in the random hypergraph $2$-coloring problem.