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.