Monotonicity of Avoidance Coupling on $K_N$

Ohad Noy Feldheim

Published 2014-10-19Version 1

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. To do this we introduce a new generalization of simple avoidance coupling which we call partially-ordered simple avoidance coupling.