Comparison of limit shapes for Bernoulli first-passage percolation

Abstract

We consider Bernoulli first-passage percolation on the ddimensional 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 pc be the critical probability for percolation of edges with passage time 0. When 0 ≤ p < pc, there exists a nonrandom, nonempty compact convex set Bp such that the set of vertices to which the firstpassage time from the origin is within t is well-approximated by tBp for all large t, with probability one. The aim of this paper is to prove that for 0 ≤ p < q < pc, the Hausdorff distance between Bp and Bq 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.