Comparison of limit shapes for Bernoulli first-passage percolation

Abstract

We consider Bernoulli first-passage percolation on the $d$-dimensional hypercubic lattice with $d\ge 2$. 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 $0\le p<p_c$, there exists a nonrandom, nonempty compact convex set ${\mathcal{B}}_p$ such that the set of vertices to which the first-passage time from the origin is within $t$ is well-approximated by $t{\mathcal{B}}_p$ for all large $t$, with probability one. The aim of this paper is to prove that for $0\le p<q<p_c$, the Hausdorff distance between ${\mathcal{B}}_p$ and ${\mathcal{B}}_q$ 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 \textit{critical} case.

9 pages