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 $d\ge 10$ and $p=1/2$, all sequences can be embedded, almost surely. They conjectured that the same should hold for $d\ge 3$. In this paper we consider $d\ge 3$ 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.

arXiv comment13 pages, 5 figures

References in corpus (1)

• Lipschitz embeddings of random sequences