Random walks on regular trees can not be slowed down

Abstract

We study a permuted random walk process on a graph G. Given a fixed sequence of permutations on the vertices of G, the permuted random walker alternates between taking random walk steps, and applying the next permutation in the sequence to their current position. Existing work on permuted random walks includes results on hitting times, mixing times, and asymptotic speed. The usual random walk on a regular tree, or generally any non-amenable graph, has positive speed, i.e. the distance from the origin grows linearly. Our focus is understanding whether permuted walks can be slower than the corresponding non-permuted walk, by carefully choosing the permutation sequence. We show that on regular trees (including the line), the permuted random walk is always stochastically faster. The proof relies on a majorization inequality for probability measures, plus an isoperimetric inequality for the tree. We also quantify how much slower the permuted random walk can possibly be when it is coupled with the corresponding non-permuted walk.