The weak Bruhat order for random walks on Coxeter groups

TL;DR

This paper proves that for simple random walks on Coxeter groups, the likelihoods of states follow the weak Bruhat order, with the identity being most likely and the longest element least likely, under certain conditions.

Contribution

It establishes that the likelihood distribution of states in a random walk on Coxeter groups respects the weak Bruhat order, extending to cases with non-uniform generator probabilities.

Findings

Likelihoods decrease along geodesics from the identity

Most likely element is the identity

Least likely element is the longest element in finite groups

Abstract

We show that for the simple random walk on a Coxeter group generated by the Coxeter generators and identity, the likelihoods of being at any pair of states respect the weak Bruhat order. That is, after any number of steps, the most likely element is the identity, probabilities decrease along any geodesic from the identity, and the least likely element is the longest element, if the group is finite. The result remains true when different generators have different probabilities, so long as the identity is at least as likely as any other.