The weak Bruhat order for random walks on Coxeter groups

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.