Dobrushin conditions for systematic scan with block dynamics

Abstract

We study the mixing time of systematic scan Markov chains on finite spin systems. It is known that, in a single site setting, the mixing time of systematic scan can be bounded in terms of the influences sites have on each other. We generalise this technique for bounding the mixing time of systematic scan to block dynamics, a setting in which a set of sites are updated simultaneously. In particular we present a parameter α, representing the maximum influence on any site, and show that if α < 1 then the corresponding systematic scan Markov chain mixes rapidly. We use this method to prove O(log n) mixing of a systematic scan for proper q-colourings of a general graph with maximum vertex-degree Δ whenever q ≥ 2Δ. We also apply the method to improve the number of colours required in order to obtain mixing in O(log n) scans for a systematic scan colouring of trees. © Springer-Verlag Berlin Heidelberg 2007.