An extension of path coupling and its application to the Glauber dynamics for graph colourings (Extended abstract)

A new method for analysing the mixing time of Markov chains is described. This method is an extension of path coupling and involves analysing the coupling over multiple steps. The expected behaviour of the coupling at a certain stopping time is used to bound the expected behaviour of the coupling after a fixed number of steps. We apply the new method to analyse the mixing time of the Glauber dynamics for graph colourings. We show that the Glauber dynamics has O(n log(n)) mixing time for triangle-free Delta-regular graphs if k colours are used, where k greater than or equal to (2 - eta)Delta, for some small positive constant eta. This is the first proof of an optimal upper bound for the mixing time of the Glauber dynamics in the range k less than or equal to 2 Delta.