Phase transition in the family of p-resistances

We study the family of p-resistances on graphs for p 1. We prove that for any fixed graph, for p 1, the p-resistance coincides with the shortest path distance, for p 2 it coincides with the standard resistance distance, and for p it converges to the inverse of the minimal s-t-cut in the graph. Secondly, we consider the special case of random geometric graphs (such as k-nearest neighbor graphs) when the number n of vertices in the graph tends to infinity. We prove that an interesting phase-transition takes place. There exist two critical thresholds p * and p * such that if p p, then the p-resistance depends on meaningful global properties of the graph, whereas if p p, it only depends on trivial local quantities and does not convey any useful information. We can explicitly compute the critical values: p * 1 1/(d-1) and p 1 1/(d-2) where d is the dimension of the underlying space (we believe that the fact that there is a small gap between p * and p * is an artifact of our proofs.