Paper detail

The travel time in a finite box in supercritical Bernoulli percolation

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