Hahn polynomials and the Burnside process

TL;DR

This paper analyzes a Markov chain called the Burnside process, whose eigenvectors are Hahn polynomials, providing explicit convergence rates and generalizations related to the Swendsen-Wang algorithm and permutation models.

Contribution

It introduces a new analysis of the Burnside process using Hahn polynomials, enabling sharp convergence rate calculations and a useful generalization of the process.

Findings

Explicit diagonalization of the Markov chain using Hahn polynomials

Sharp convergence rates to stationarity derived

Generalization involving beta-binomial distribution and permutation models

Abstract

We study a natural Markov chain on $\{0,1,\cdots,n\}$ 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.