Recurrence or transience of random walks on random graphs generated by point processes in RdRd

We consider random walks associated with conductances on Delaunay triangulations, Gabriel graphs and skeletons of Voronoi tilings generated by point processes in RdRd. 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=2d=2 and transient if d≥3d≥3. These results hold for a large variety of point processes including Poisson point processes, Matérn cluster and Matérn hardcore processes which have clustering or repulsive properties. In order to prove them, we state general criteria for recurrence or transience which apply to random graphs embedded in RdRd.