Monotonicity of Avoidance Coupling on KN

Abstract

Answering a question by Angel, Holroyd, Martin, Wilson and Winkler [1], we show that the maximal number of non-colliding coupled simple random walks on the complete graph KN, which take turns, moving one at a time, is monotone in N. We use this fact to couple [N/4] such walks on KN, improving the previous Ω(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.