The random $k$ cycle walk on the symmetric group

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 $\frac{n}{k}logn$ 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.

This paper first appeared as part of the author's PhD thesis