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Mixing times of a Burnside process Markov chain on set partitions

J. E. Paguyo

TL;DR

This work analyzes the Burnside process on $[k]^n$ with $S_k$ acting by permuting coordinate values, showing that the induced chain on set partitions $\\\\ $ mixes rapidly in two regimes: $k\ge n$ and $1\le k<n$. Using coupling and minorization, the authors establish explicit mixing-time bounds: in the first regime, $t_{mix}(\\varepsilon)\le\\lceil 2k\\log(n/\\varepsilon)\\rceil$, and in the second, $t_{mix}(\\varepsilon)\le\\lceil (k-1)! \\log(1/\\varepsilon)\\rceil$, with independence from $n$ for fixed $k$. They also derive explicit transition probabilities both for the original process on $[k]^n$ and its lumped chain on partitions, and obtain eigenvalue bounds for the lumped and original chains, linking spectral properties to mixing via relaxation times. These results provide a rigorous foundation for using the Burnside process to uniformly sample set partitions and offer insights into the chain’s spectral structure, while highlighting open problems in diagonalizing the transition operators and proving sharp lower bounds. The work thus contributes practical mixing-time guarantees for an orbit-based sampler and connects combinatorial structure with Markov-chain techniques such as coupling, minorization, and geometric bounds.

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.

Mixing times of a Burnside process Markov chain on set partitions

TL;DR

This work analyzes the Burnside process on with acting by permuting coordinate values, showing that the induced chain on set partitions mixes rapidly in two regimes: and . Using coupling and minorization, the authors establish explicit mixing-time bounds: in the first regime, , and in the second, , with independence from for fixed . They also derive explicit transition probabilities both for the original process on and its lumped chain on partitions, and obtain eigenvalue bounds for the lumped and original chains, linking spectral properties to mixing via relaxation times. These results provide a rigorous foundation for using the Burnside process to uniformly sample set partitions and offer insights into the chain’s spectral structure, while highlighting open problems in diagonalizing the transition operators and proving sharp lower bounds. The work thus contributes practical mixing-time guarantees for an orbit-based sampler and connects combinatorial structure with Markov-chain techniques such as coupling, minorization, and geometric bounds.

Abstract

Let be a finite set and let be a finite group acting on . The group action splits into disjoint orbits. The Burnside process is a Markov chain on which has a uniform stationary distribution when the chain is lumped to orbits. We consider the case where with and is the symmetric group on , such that acts on by permuting the value of each coordinate. The resulting Burnside process gives a novel algorithm for sampling a set partition of uniformly at random. We obtain bounds on the mixing time and show that the chain is rapidly mixing. For the case , the algorithm corresponds to sampling a set partition of with at most blocks, and we obtain a mixing time bound which is independent of . 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.
Paper Structure (17 sections, 17 theorems, 58 equations)

This paper contains 17 sections, 17 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. The Burnside Process
  3. Main Results
  4. Outline
  5. Preliminaries
  6. Markov Chains and Mixing Times
  7. Auxiliary Variables
  8. Coupling
  9. Minorization
  10. Spectral Gap
  11. Geometric Bounds
  12. Transition Probabilities
  13. Mixing Time Upper Bounds
  14. Coupling Bound
  15. Minorization Bound
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $K(u,v)$ be the transition matrix of the Burnside process on $[k]^n$, where $k \geq n \geq 1$. Let $\pi(u) = \frac{1}{B_n |O_u|}$ be its stationary distribution. Then The upper bound is uniform in the starting state $u \in [k]^n$. Therefore the mixing time of $K$ satisfies

Theorems & Definitions (26)

  • Theorem 1.1
  • Proposition 2.1: Ros95a, Proposition 2
  • Proposition 2.2: Sal97, Theorem 2.1.7
  • Proposition 2.3: DS91, Proposition 1'
  • Proposition 2.4: DS91, Proposition 6
  • Proposition 2.5: LPW09, Theorems 12.4 and 12.5
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 4.1: AF02, Lemma 12.4
  • ...and 16 more