On the Meeting Time for Two Random Walks on a Regular Graph

TL;DR

This paper analyzes the expected meeting time of two independent random walks on regular graphs, providing exact formulas for specific cases and conjecturing broader applicability, with results showing quadratic and near-quadratic growth rates.

Contribution

It derives explicit formulas for meeting times on 1-D and 2-D regular graphs and conjectures their generalization to all regular graphs.

Findings

Expected meeting time on 1-D circle is Θ(N^2).

Expected meeting time on 2-D torus is Θ(N^2 log N).

Meeting time relates to eigenvalues of Laplacian matrices.

Abstract

We provide an analysis of the expected meeting time of two independent random walks on a regular graph. For 1-D circle and 2-D torus graphs, we show that the expected meeting time can be expressed as the sum of the inverse of non-zero eigenvalues of a suitably defined Laplacian matrix. We also conjecture based on empirical evidence that this result holds more generally for simple random walks on arbitrary regular graphs. Further, we show that the expected meeting time for the 1-D circle of size $N$ is $\Theta(N^2)$, and for a 2-D $N \times N$ torus it is $\Theta(N^2 log N)$.