Comparison of limit shapes for Bernoulli first-passage percolation

Abstract

. We consider Bernoulli first-passage percolation on the d dimensional hypercubic lattice with d ≥ 2. The passage time of edge e is 0 with probability p and 1 with probability 1 − p , independently of each other. Let p c be the critical probability for percolation of edges with passage time 0. When 0 ≤ p < p c , there exists a nonrandom, nonempty compact convex set B p such that the set of vertices to which the first-passage time from the origin is within t is well-approximated by t B p for all large t , with probability one. The aim of this paper is to prove that for 0 ≤ p < q < p c , the Hausdorff distance between B p and B q grows linearly in q − p . Moreover, we mention that the approach taken in the paper provides a lower bound for the expected size of the intersection of geodesics, that gives a nontrivial consequence for the critical case.