The travel time in a finite box in supercritical Bernoulli percolation

TL;DR

This paper proves that in supercritical Bernoulli percolation on a 3D lattice, most vertex pairs in a finite box are connected by paths with a logarithmic squared number of closed sites, highlighting efficient connectivity.

Contribution

It establishes a probabilistic bound on the travel time between vertices in a finite box under supercritical percolation, assuming only positive percolation probability.

Findings

High probability of short paths with few closed sites between vertices

Bound on the number of closed sites grows as (\ln n)^2

Connectivity properties hold for large finite boxes

Abstract

We consider the standard site percolation model on the three dimensional cubic lattice. Starting solely with the hypothesis that $\theta(p)>0$, we prove that, for any $\alpha>0$, there exists $\kappa>0$ such that, with probability larger than $1-1/n^\alpha$, every pair of vertices inside the box $\Lambda(n)$ are joined by a path having at most $\kappa(\ln n)^2$ closed sites.