I'll introduce the Burnside process: a family of finite state-space Markov chains that have proved surprisingly effective to simulate things such as contingency tables with fixed row and column sums, partitions of n (when n is 10^8) or more general objects (trees, graphs) defined up to symmetry. The chains seem to converge lightning-fast, in less than 100 steps for partitions of n=10^6. Unfortunately, it has been very hard to prove anything about their mixing time.
In joint work with Andrew Lin and Arun Ram, we have gotten sharp answers for the "simplest" case. The Binary Burnside process runs on binary n-tuples. Started from 0, it converges in a bounded number of steps, no matter how large n is. From any start, it converges in order log n steps, in l_1; but for most starts, order n/log n steps are necessary and sufficient in the chi-square distance. The arguments make nice use of Schur–Weyl duality with sl(2,C) and make good examples of spectral analysis techniques of various sorts. We hope they will help with "real" problems.