Scheduling of non-colliding random walks

TL;DR

This paper proves Winkler's conjecture that for sufficiently large complete graphs, there exists a positive measure set of non-colliding random walk pairs, by linking the problem to the existence of an infinite open cluster in a dependent percolation model.

Contribution

The paper confirms Winkler's conjecture by demonstrating the existence of non-colliding random walk pairs for large graphs through percolation theory.

Findings

Confirmed Winkler's conjecture for large M

Established the existence of an infinite open cluster in the related percolation model

Connected non-colliding walks problem to percolation theory

Abstract

On the complete graph ${\cal{K}}_M$ with $M \ge3$ vertices consider two independent discrete time random walks $\mathbb{X}$ and $\mathbb{Y}$, choosing their steps uniformly at random. A pair of trajectories $\mathbb{X} = \{ X_1, X_2, \dots \}$ and $\mathbb{Y} = \{Y_1, Y_2, \dots \}$ is called {\it{non-colliding}}, if by delaying their jump times one can keep both walks at distinct vertices forever. It was conjectured by P. Winkler that for large enough $M$ the set of pairs of non-colliding trajectories $\{\mathbb{X},\mathbb{Y} \} $ has positive measure. N. Alon translated this problem to the language of coordinate percolation, a class of dependent percolation models, which in most situations is not tractable by methods of Bernoulli percolation. In this representation Winkler's conjecture is equivalent to the existence of an infinite open cluster for large enough $M$. In this paper we establish the conjecture.