Comparison of limit shapes for Bernoulli first-passage percolation

TL;DR

This paper investigates how the limit shapes in Bernoulli first-passage percolation on hypercubic lattices change as the probability parameter varies below the critical threshold, establishing a linear growth in shape differences.

Contribution

It proves that the Hausdorff distance between limit shapes for different probabilities below criticality grows linearly with the difference in probabilities.

Findings

Hausdorff distance between shapes grows linearly with probability difference

Provides a lower bound for the expected intersection size of geodesics

Offers insights into the critical case behavior

Abstract

We consider Bernoulli first-passage percolation on the $d$-dimensional hypercubic lattice with $d \geq 2$. 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 $0\leq p<p_c$, 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 $0\leq p<q<p_c$, 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.