The travel time in a finite box in supercritical Bernoulli percolation

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.