Hahn polynomials and the Burnside process

We study a natural Markov chain on $$\{0,1,\ldots ,n\}$$ with eigenvectors the Hahn polynomials. This explicit diagonalization makes it possible to get sharp rates of convergence to stationarity. The process, the Burnside process, is a special case of the celebrated ‘Swendsen–Wang’ or ‘data augmentation’ algorithm. The description involves the beta-binomial distribution and Mallows model on permutations. It introduces a useful generalization of the Burnside process.