No exceptional words for Bernoulli percolation

Abstract

Benjamini and Kesten introduced in 1995 the problem of embedding infinite binary sequences into a Bernoulli percolation configuration, known as percolation of words. We give a positive answer to their Open Problem 2: almost surely, all words are seen for site percolation on ${\mathbb{Z}}^3$ with parameter $p=1/2$. We also extend this result in various directions, proving the same result on ${\mathbb{Z}}^d$, $d\ge 3$, for any value $p\in (p_c^{\text{site}}({\mathbb{Z}}^d),1-p_c^{\text{site}}({\mathbb{Z}}^d))$, and for restrictions to slabs. Finally, we provide an explicit estimate on the probability to find all words starting from a finite box.