The ‘Burnside Process’ Converges Slowly

We consider the problem of sampling ‘unlabelled structures’, i.e., sampling combinatorial structures modulo a group of symmetries. The main tool which has been used for this sampling problem is Burnside’s lemma. In situations where a significant proportion of the structures have no nontrivial symmetries, it is already fairly well understood how to apply this tool. More generally, it is possible to obtain nearly uniform samples by simulating a Markov chain that we call the Burnside process: this is a random walk on a bipartite graph which essentially implements Burnside’s lemma. For this approach to be feasible, the Markov chain ought to be ‘rapidly mixing’, i.e., converge rapidly to equilibrium. The Burnside process was known to be rapidly mixing for some special groups, and it has even been implemented in some computational group theory algorithms. In this paper, we show that the Burnside process is not rapidly mixing in general. In particular, we construct an infinite family of permutation groups for which we show that the mixing time is exponential in the degree of the group.