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 $\mathbb{Z}^d$, $d \geq 3$, for any value $p \in (p\_c^{\textup{site}}(\mathbb{Z}^d), 1 - p\_c^{\textup{site}}(\mathbb{Z}^d))$, and for restrictions to slabs. Finally, we provide an explicit estimate on the probability to find all words starting from a finite box.