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

TL;DR

This work analyzes the mixing time of the Sylow--Burnside process on $S_{pk}$ with a Sylow $p$-subgroup $H$ under the action $\sigma^{h,g}=h^{-1}\sigma g$, in the Abelian regime $n=pk$, $k<p$. It introduces a lumping technique based on the double coset size $|H\sigma H|=p^a$ and proves that most double cosets are large, yielding a reduced Markov chain on the set $\{k,\dots,2k\}$ with stationary distribution $\overline{\pi}(a)=f(a;k)/Z$. The main results give a sharp limit profile for the worst-case total-variation distance: if $k$ is fixed, the chain mixes in $t\approx cp$ steps with no cutoff, while if $k\to\infty$, a cutoff occurs at $t\approx p\log k$ with window $p$, with explicit formulas $d(cp)\to 1-(1-e^{-c})^k$ and $d(p\log k+cp)\to 1-\exp(-e^{-c})$. The analysis provides explicit, non-asymptotic bounds that remain accurate even for small primes (e.g., $p=11$) and offers a practical path to sampling Sylow $p$-double cosets efficiently. These results advance the understanding of Burnside-process mixings in natural group actions and illustrate a powerful lumping strategy that may extend to non-Abelian settings.

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.

Figures (1)

• Figure 1: Both figures show a plot of $\Vert \hat{\mu}_t - \overline{\pi}\Vert_{\mathrm{TV}}$ as a function of $t$ for different values of $p$ and $k$. In both cases, the empirical measure $\hat{\mu}_t$ is estimated based on $B=10,000$ runs of the Sylow--Burnside process. The plots also show the limit profile $t \mapsto 1-\exp(-e^{-c})$ where $c = (t-p\log k)/p$ from Theorem \ref{['thrm:mix']} and the approximation $t \mapsto 1-(1-(1-1/p)^t)^k$ from Theorem \ref{['thrm:bound']}. Both approximations are accurate when $p=23$ and $k=22$ but only Theorem \ref{['thrm:bound']} is accurate for $p=11$ and $k=10$. The high accuracy of Theorem \ref{['thrm:bound']} is reflected in the fact that the error term goes to zero super-exponential fast in $p$. The code use to create Figure \ref{['fig:sim']} is available at https://github.com/Michael-Howes/BurnsideProcess/blob/main/Examples/sylow_double_cosets_example.ipynb

Theorems & Definitions (46)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 2.1: Theorem 3.1 in diaconis2025number
• Theorem 2.2: Theorem 3.3 in diaconis2025number
• Proposition 2.3
• Lemma 3.1