Uniform Avoidance Coupling of Simple Random Walks

TL;DR

This paper introduces the concept of uniform avoidance coupling for simple random walks, explores its existence on various graphs, and provides algorithms to test for such couplings.

Contribution

It defines uniform avoidance coupling, characterizes its existence on different graph classes, and offers a polynomial-time algorithm to test for it.

Findings

Uniform avoidance coupling is only possible on cycles for Markovian cases.

Impossible on trees for simple random walks.

Possible on several other graph classes.

Abstract

We start by introducing avoidance coupling of Markov chains, with an overview of existing results. We then introduce and motivate a new notion, uniform avoidance coupling. We show that the only Markovian avoidance coupling on a cycle is of this type, and that uniform avoidance coupling of simple random walks is impossible on trees, and prove that it is possible on several classes of graphs. We also derive a condition on the vertex neighborhoods in a graph equivalent to that graph admitting a uniform avoidance coupling of simple random walks, and an algorithm that tests this with run time polynomial in the number of vertices.