Monotonicity of Avoidance Coupling on $K_N$

TL;DR

This paper proves the monotonicity of the maximum number of non-colliding simple random walks on complete graphs and introduces a new generalized avoidance coupling with similar properties.

Contribution

It establishes the monotonicity of avoidance coupling on complete graphs and introduces a new partially ordered avoidance coupling extension.

Findings

Maximal non-colliding walks are monotone in N

Couples eil; N/4  walks on K_N achieved

New partial order avoidance coupling introduced

Abstract

Answering a question by Angel, Holroyd, Martin, Wilson and Winkler, we show that the maximal number of non-colliding coupled simple random walks on the complete graph $K_N$, which take turns, moving one at a time, is monotone in $N$. We use this fact to couple $\lceil \frac N4 \rceil$ such walks on $K_N$, improving the previous $\Omega(N/\log N)$ lower bound of Angel et al. We also introduce a new generalization of simple avoidance coupling which we call partially ordered simple avoidance coupling and provide a monotonicity result for this extension as well.