Sampling 3-colourings of regular bipartite graphs
David J Galvin, University of Pennsylvania
Abstract
We show that if G=(V,E) is a regular bipartite graph for which the expansion of subsets of a single parity of V is reasonably good and which satisfies a certain local condition (that the union of the neighbourhoods of adjacent vertices does not contain too many pairwise non-adjacent vertices), and if M is a Markov chain on the set of proper 3-colourings of G which updates the colour of at most c|V| vertices at each step and whose stationary distribution is uniform, then for c < .22 and d sufficiently large the convergence to stationarity of M is (essentially) exponential in |V|. In particular, if G is the d-dimensional hypercube Q_d (the graph on vertex set {0,1}^d in which two strings are adjacent if they differ on exactly one coordinate) then the convergence to stationarity of the well-known Glauber (single-site update) dynamics is exponentially slow in 2^d/(log d d^(1/2)). A combinatorial corollary of our main result is that in a uniform 3-colouring of Q_d there is an exponentially small probability (in 2^d) that there is a colour i such the proportion of vertices of the even subcube coloured i differs from the proportion of the odd subcube coloured i by at most .22. Our proof combines a conductance argument with combinatorial enumeration methods.
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