Random walks on regular trees can not be slowed down

TL;DR

This paper proves that on regular trees, a random walk's positive speed cannot be reduced by any time-dependent permutations of vertices, highlighting inherent limitations in controlling such stochastic processes.

Contribution

It establishes that random walks on regular trees cannot be slowed down through vertex permutations, a novel result in the study of non-amenable graphs.

Findings

Random walks on regular trees have positive speed.

Vertex permutations cannot slow down these walks.

The result applies to non-amenable graphs generally.

Abstract

A random walk on a regular tree (or any non-amenable graph) has positive speed. We ask whether such a walk can be slowed down by applying carefully chosen time-dependent permutations of the vertices. We prove that on trees the random walk can not be slowed down.