The random (n-k)-cycle to transpositions walk on the symmetric group

TL;DR

This paper analyzes the convergence rate of a Markov chain on the symmetric group starting from an (n-k)-cycle and then applying random transpositions, showing it converges in order n steps with detailed bounds.

Contribution

It provides new bounds and asymptotic analysis for the convergence of a specific Markov chain on the symmetric group starting from an (n-k)-cycle.

Findings

Convergence occurs around order n steps.

Lower bounds established using the character of the defining representation.

Asymptotic distribution identified for the (n-1)-cycle case.

Abstract

We study the rate of convergence of the Markov chain on $S_n$ which starts with a random $(n-k)$-cycle for a fixed $k \geq 1$, followed by random transpositions. The convergence to the stationary distribution turns out to be of order $n$. We show that after $cn + \frac{\ln k}{2}n$ steps for $c>0$, the law of the Markov chain is close to the uniform distribution. The character of the defining representation is used as test function to obtain a lower bound for the total variation distance. We identify the asymptotic distribution of the test function given the law of the Markov chain for the $(n-1)$-cycle case. The upper bound relies on estimates for the difference of normalized characters.