Path Coupling Using Stopping Times and Counting Independent Sets and Colourings in Hypergraphs

We analyse the mixing time of Markov chains 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.