Precise asymptotics of some meeting times arising from the voter model on large random regular graphs

TL;DR

This paper derives precise asymptotic formulas for the expected meeting times of two independent stationary random walks on large random regular graphs, refining existing exponential approximations and aiding voter model diffusion analysis.

Contribution

It provides explicit asymptotic expressions for meeting times, improving the accuracy of previous exponential approximations in the context of large random regular graphs.

Findings

Expected meeting time asymptotics: N(k-1)/[2(k-2)]

Refined exponential approximation for meeting times

Enhanced understanding of voter model diffusion processes

Abstract

We consider two independent stationary random walks on large random regular graphs of degree $k\geq 3$ with $N$ vertices. On these graphs, the exponential approximations of the meeting times are known to follow from existing methods and form a basis for the voter model's diffusion approximations. The main result of this note improves the exponential approximations to an explicit form such that the first moments are asymptotically equivalent to $N(k-1)/[2(k-2)]$.