The random $k$ cycle walk on the symmetric group

TL;DR

This paper analyzes the mixing time of a random walk on the symmetric group generated by k-cycles, revealing a cutoff phenomenon at approximately (n/k) log n steps, based on new character estimates.

Contribution

It introduces a novel asymptotic analysis of symmetric group characters to determine the cutoff for k-cycle walks.

Findings

The walk exhibits a cutoff at (n/k) log n steps.

Convergence to uniform is uniform for k=o(n).

New character asymptotics underpin the analysis.

Abstract

We study the random walk on the symmetric group $S_n$ generated by the conjugacy class of cycles of length $k$. We show that the convergence to uniform measure of this walk has a cut-off in total variation distance after $\frac{n}{k} log n$ steps, uniformly in $k = o(n)$ as $n \to \infty$. The analysis follows from a new asymptotic estimation of the characters of the symmetric group evaluated at cycles.