Comparison of limit shapes for Bernoulli first-passage percolation

Abstract

We consider Bernoulli first-passage percolation on the [Formula: see text]-dimensional hypercubic lattice with [Formula: see text]. The passage time of edge [Formula: see text] is 0 with probability [Formula: see text] and 1 with probability [Formula: see text], independently of each other. Let [Formula: see text] be the critical probability for percolation of edges with passage time 0. When [Formula: see text], there exists a nonrandom, nonempty compact convex set [Formula: see text] such that the set of vertices to which the first-passage time from the origin is within [Formula: see text] is well approximated by [Formula: see text] for all large [Formula: see text], with probability one. The aim of this paper is to prove that for [Formula: see text], the Hausdorff distance between [Formula: see text] and [Formula: see text] grows linearly in [Formula: see text]. 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.