In this talk we consider a class of random simple point processes and the random walk under random conductances on the associated Delaunay triangulation. We present comparison results between balls in the graph distance on the Delaunay triangulation (a.k.a. the chemical
distance) and the Euclidean distance. Using this result we derive a quenched invariance principle for the random walk under suitable ergodicity and moment assumptions on both, the point process and the conductances.
This task is based on a joint work with Alessandra Faggionato, Martin Slowik and Yuki Tokushige.