Recurrence or transience of random walks on random graphs generated by point processes in $\mathbb{R}^d$

TL;DR

This paper investigates the recurrence or transience of random walks on various random geometric graphs generated by point processes in Euclidean space, establishing dimension-dependent behavior under broad conditions.

Contribution

It provides general criteria for recurrence and transience of random walks on random geometric graphs generated by diverse point processes in R^d, covering many common models.

Findings

Random walks are recurrent in 2D and transient in 3D or higher.

Results apply to Poisson, Me9te9rn cluster, and hardcore processes.

Criteria for recurrence/transience are established for embedded random graphs.

Abstract

We consider random walks associated with conductances on Delaunay triangulations, Gabriel graphs and skeletons of Voronoi tilings which are generated by point processes in $\mathbb{R}^d$. Under suitable assumptions on point processes and conductances, we show that, for almost any realization of the point process, these random walks are recurrent if $d=2$ and transient if $d\geq 3$. These results hold for a large variety of point processes including Poisson point processes, Mat{\'e}rn cluster and Mat{\'e}rn hardcore processes which have clustering or repulsive properties. In order to prove them, we state general criteria for recurrence or almost sure transience which apply to random graphs embedded in $\mathbb{R}^d$.