Embedding binary sequences into Bernoulli site percolation on $\mathbb{Z}^3$

Abstract: We investigate the problem of embedding infinite binary sequences into Bernoulli site percolation on $\mathbb{Z}^d$ with parameter $p$, known also as percolation of words.\ In 1995, I.\ Benjamini and H.\ Kesten proved that, for $d \geq 10$ and $p=1/2$, all sequences can be embedded, almost surely. They conjectured that the same should hold for $d \geq 3$. In this paper we consider $d \geq 3$ and $p \in (p_c(d), 1-p_c(d))$, where $p_c(d)<1/2$ is the critical threshold for site percolation on $\mathbb{Z}^d$. We show that there exists an integer $M = M (p)$, such that, a.s., every binary sequence, for which every run of consecutive {0s} or {1s} contains at least $M$ digits, can be embedded.