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

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 $\mathbb{R}^d$.