Mixing times of a Burnside process Markov chain on set partitions

TL;DR

This paper analyzes the mixing times of a Burnside process Markov chain on set partitions, providing bounds and explicit formulas, and introduces a novel sampling algorithm with rapid mixing properties.

Contribution

It introduces a new Burnside process-based algorithm for sampling set partitions uniformly and establishes bounds on its rapid mixing behavior.

Findings

The chain is proven to be rapidly mixing for certain parameters.

Explicit formulas for transition probabilities are derived.

Mixing time bounds are obtained, independent of n when k < n.

Abstract

Let $X$ be a finite set and let $G$ be a finite group acting on $X$. The group action splits $X$ into disjoint orbits. The Burnside process is a Markov chain on $X$ which has a uniform stationary distribution when the chain is lumped to orbits. We consider the case where $X = [k]^n$ with $k \geq n$ and $G = S_k$ is the symmetric group on $[k]$, such that $G$ acts on $X$ by permuting the value of each coordinate. The resulting Burnside process gives a novel algorithm for sampling a set partition of $[n]$ uniformly at random. We obtain bounds on the mixing time and show that the chain is rapidly mixing. For the case $k < n$, the algorithm corresponds to sampling a set partition of $[n]$ with at most $k$ blocks, and we obtain a mixing time bound which is independent of $n$. Along the way, we obtain explicit formulas for the transition probabilities and bounds on the second largest eigenvalue for both the original process and the lumped chain.