Path coupling using stopping times and counting independent sets and colorings in hypergraphs

We give a new method for analysing the mixing time of a Markov chain using path coupling with stopping times. We apply this approach to two hypergraph problems. We show that the Glauber dynamics for independent sets in a hypergraph mixes rapidly as long as the maximum degree ∆ of a vertex and the minimum size m of an edge satisfy m ≥ 2∆ + 1. We also show that the Glauber dynamics for proper q-colourings of a hypergraph mixes rapidly if m ≥ 4 and q > ∆, and if m = 3 and q ≥ 1.65∆. We give related results on the hardness of exact and approximate counting for both problems.