The condensation phase transition in random graph coloring

TL;DR

This paper rigorously proves the existence and precise location of the condensation phase transition in random graph k-coloring for sufficiently large k, confirming longstanding physics-based predictions.

Contribution

It provides a rigorous proof of the conjectured condensation phase transition in random graph k-coloring for large k, bridging physics predictions and mathematical validation.

Findings

Confirmed the existence of the condensation phase transition.

Precisely located the transition point for large k.

Linked the transition to computational difficulty and algorithm performance.

Abstract

Based on a non-rigorous formalism called the "cavity method", physicists have put forward intriguing predictions on phase transitions in discrete structures. One of the most remarkable ones is that in problems such as random $k$-SAT or random graph $k$-coloring, very shortly before the threshold for the existence of solutions there occurs another phase transition called "condensation" [Krzakala et al., PNAS 2007]. The existence of this phase transition appears to be intimately related to the difficulty of proving precise results on, e.g., the $k$-colorability threshold as well as to the performance of message passing algorithms. In random graph $k$-coloring, there is a precise conjecture as to the location of the condensation phase transition in terms of a distributional fixed point problem. In this paper we prove this conjecture for $k$ exceeding a certain constant $k_0$.