Rapid mixing of Glauber dynamics at the ergodicity threshold

Prove that the heat-bath Glauber dynamics for uniformly random proper vertex q-colorings mixes rapidly on every finite graph of maximum degree Δ whenever q≥Δ+2, thereby establishing the long-standing folklore conjecture for the full irreducible regime.

Background

The paper studies the efficiency of heat-bath Glauber dynamics for sampling proper q-colorings. The chain is ergodic when q≥Δ+2, while the paper proves rapid mixing only for graphs of girth at least five in the more restrictive asymptotic regime q≥(1+δ)Δ, with sufficiently large Δ depending on δ.

The conjecture concerns arbitrary graphs and the entire threshold range q≥Δ+2. Thus it remains substantially broader than the girth-five results established in the paper and represents the unresolved target motivating the surrounding discussion of prior work.