Mixing time and isoperimetry in random geometric graphs

TL;DR

This paper analyzes the mixing time of random walks on the giant component of supercritical random geometric graphs, establishing precise asymptotics using isoperimetric inequalities, and closes a gap in existing literature.

Contribution

It provides a sharp characterization of mixing times for the giant component in supercritical random geometric graphs, using novel isoperimetric inequalities.

Findings

Mixing time is Θ(n^{2/d}/r^{2}) for r above the percolation threshold.

Isoperimetric inequality holds for large vertex sets with high probability.

Relaxation time is of the same order as mixing time.

Abstract

In this paper we study the mixing time of the simple random walk on the giant component of supercritical $d$-dimensional random geometric graphs generated by the unit intensity Poisson Point Process in a $d$-dimensional cube of volume $n$. With $r_g$ denoting the threshold for having a giant component, we show that for every $\epsilon > 0$ and any $r \ge (1+\epsilon)r_g$, the mixing time of the giant component is with high probability $\Theta(n^{2/d}/r^{2})$, thereby closing a gap in the literature. The main tool is an isoperimetric inequality which holds, w.h.p., for any large enough vertex set, a result which we believe is of independent interest. Our analysis also implies that the relaxation time is of the same order.