Mixing times of a Burnside process Markov chain on set partitions
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915747625697280 |
|---|---|
| author | Paguyo, J. E. |
| author_facet | Paguyo, J. E. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_14269 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Mixing times of a Burnside process Markov chain on set partitions Paguyo, J. E. Probability Combinatorics 60J10, 60C05 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. |
| title | Mixing times of a Burnside process Markov chain on set partitions |
| topic | Probability Combinatorics 60J10, 60C05 |
| url | https://arxiv.org/abs/2207.14269 |