Limit profiles and cutoff for the Burnside process on Sylow double cosets

TL;DR

This paper analyzes the mixing time and cutoff phenomena of the Burnside process on Sylow p-double cosets in symmetric groups, providing sharp estimates and limit profiles as parameters grow.

Contribution

It offers the first detailed analysis of the Burnside process's cutoff behavior on Sylow double cosets, including explicit bounds and asymptotic profiles.

Findings

Order p steps are necessary and sufficient for mixing when k is fixed.

Cutoff occurs at p log k when k grows with p.

Explicit bounds give accurate estimates even for small p.

Abstract

This article gives sharp estimates for the mixing time of the Burnside process for Sylow $p$-double cosets in the symmetric group $S_n$. This process is a Markov chain on $S_n$ which can be used to uniformly sample Sylow $p$-double cosets. The analysis applies when $n = pk$ with $p$ prime and $k < p$. The main result describes the limit profile of the distance to the stationary distribution as $p$ goes to infinity. From the limit profile, we get the following two corollaries. First, if $k$ remains fixed as $p \to \infty$, then order $p$ steps are necessary and sufficient for mixing and cut-off does not occur. Second, if $k \to \infty$ as $p \to \infty$, then cut-off occurs at $p \log k$ with a window of size $p$. The limit profile is derived from explicit upper and lower bounds on the distance between the Burnside process and its stationary distribution. These non-asymptotic bounds give very accurate approximations even for $p$ as small as 11.