Mixing time of the random walk on the giant component of the random geometric graph

TL;DR

This paper analyzes the mixing time of simple random walks on the giant component of a random geometric graph in high dimensions, establishing order bounds and showing the absence of cutoff.

Contribution

It provides the first order bounds for mixing and relaxation times of the random walk on the giant component in high-dimensional random geometric graphs.

Findings

Mixing and relaxation times are both of order n^(2/d).

No cutoff phenomenon occurs for the random walk.

Bounds are established for the isoperimetric profile of large subsets.

Abstract

We consider a random geometric graph obtained by placing a Poisson point process of intensity 1 in the d-dimensional torus of side length n^(1/d) and connecting two points by an edge if their distance is at most r. We consider the case of d>=2 and r in [r_min, r_max], where r_min<r_max are any constants with r_min>r_g and r_g is a constant above which this graph has a giant component with high probability. We show that, with high probability, the mixing time and the relaxation time of the simple random walk on the giant component in this case are both of order n^(2/d) and that therefore there is no cutoff. We also obtain bounds for the isoperimetric profile of subsets of the giant component of at least polylogarithmic size.