The Glauber dynamics for colourings of bounded degree trees

Abstract. We study the Glauber dynamics Markov chain for k-colourings of trees with maximum degree ∆. For k≥3, we show that the mixing time on every tree is at most n O(1+∆/(k log ∆)). This bound is tight up to the constant factor in the exponent, as evidenced by the complete tree. Our proof uses a weighted canonical paths analysis and a variation of the block dynamics that exploits the differing relaxation times of blocks.