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

Arnaud Rousselle

Published 2013-05-21, updated 2014-09-20Version 3

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\'ern cluster and Mat\'ern 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 $\mathbb{R}^d$.