Sampling 3-colourings of regular bipartite graphs

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 $2^d/(\sqrt{d} \log d )$. 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.