Avoidance Coupling

TL;DR

This paper investigates the possibility of coupling multiple random walks on a graph to prevent collisions, demonstrating that on a complete graph, a significant number of walks can be kept apart using a Markovian coupling.

Contribution

It introduces a Markovian coupling strategy that maintains a large number of non-colliding random walks on a complete graph, advancing understanding of avoidance coupling.

Findings

Can keep Omega(n/log n) walks apart on complete graphs

Markovian coupling effectively prevents collisions

Applicable to graphs with loops and without

Abstract

We examine the question of whether a collection of random walks on a graph can be coupled so that they never collide. In particular, we show that on the complete graph on n vertices, with or without loops, there is a Markovian coupling keeping apart Omega(n/log n) random walks, taking turns to move in discrete time.