Likelihood Orders for some Random Walks on the Symmetric Group

TL;DR

This paper investigates likelihood orders in random walks on the symmetric group generated by various conjugacy classes, providing spectral analysis insights and partially confirming conjectures about the least likely elements.

Contribution

It introduces likelihood orders for specific random walks on the symmetric group and analyzes their spectral properties, advancing understanding of element likelihoods.

Findings

Likelihood orders describe element probabilities in certain random walks.

Spectral analysis determines sufficient time for likelihood orders to hold.

Partial confirmation of the conjecture that n-cycles are least likely in the transposition walk.

Abstract

Several cycle lexicographical orders are found to describe the relative likelihood of elements of the random walks on the symmetric group generated by the conjugacy classes of transpositions, 3-cycles, and n-cycles. Spectral analysis finds sufficient time for the orders to hold. This partially answers a conjecture that the n-cycles are the least likely elements of the transposition walk on the symmetric group. A likelihood order contributes to understanding the total variation distance and separation distance for a random walk.