An upper bound on the size of avoidance couplings

TL;DR

This paper establishes a new upper limit on the number of non-colliding simple random walkers on a complete graph, improving previous bounds and analyzing related Bernoulli sequence couplings.

Contribution

It provides a tighter upper bound of n - log n for avoidance couplings on complete graphs, advancing understanding of such stochastic processes.

Findings

Maximum avoidance coupling size is n - log n

Previous bound was n - 2

Analysis of Bernoulli sequence couplings related to avoidance properties

Abstract

We show that a coupling of non-colliding simple random walkers on the complete graph on $n$ vertices can include at most $n - \log n$ walkers. This improves the only previously known upper bound of $n-2$ due to Angel, Holroyd, Martin, Wilson, and Winkler ({\it Electron.~Commun.~Probab.~18}, 2013). The proof considers couplings of i.i.d.~sequences of Bernoulli random variables satisfying a similar avoidance property, for which there is separate interest. Our bound in this setting should be closer to optimal.