A random walk on the symmetric group generated by random involutions

TL;DR

This paper analyzes a new type of random walk on the symmetric group generated by involutions, providing bounds on mixing times and introducing a novel eigenvalue calculation technique using character polynomials.

Contribution

It introduces a new eigenvalue analysis method for random walks on symmetric groups generated by multiple conjugacy classes using character polynomials.

Findings

Mixing time bounds between log_{1/p}(n) and log_{2/(1+p)}(n) steps for large n

New technique for eigenvalue calculation using character polynomials

Likelihood order established after sufficient mixing time

Abstract

The involution walk is the random walk on $S_n$ generated by involutions with a binomially distributed with parameter $1-p$ number of $2$-cycles. This is a parallelization of the transposition walk. The involution walk is shown in this paper to mix for $\frac{1}{2} \leq p \leq 1$ fixed, $n$ sufficiently large in between $\log_{1/p}(n)$ steps and $\log_{2/(1+p)}(n)$ steps. The paper introduces a new technique for finding eigenvalues of random walks on the symmetric group generated by many conjugacy classes using the character polynomial for the characters of the representations of the symmetric group. Monotonicity relations used in the bound also give after sufficient time the likelihood order, the asymptotic order from most likely to least likely permutation. The walk was introduced to study a conjecture about a random walk on the unitary group from the information theory of black holes.