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

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.

Truncated Connectivities in a highly supercritical anisotropic percolation model

We consider an anisotropic bond percolation model on $mathbb{Z}^2$, with $textbf{p}=(p_h,p_v)in [0,1]^2$, $p_v>p_h$, and declare each horizontal (respectively vertical) edge of $mathbb{Z}^2$ to be open with probability $p_h$(respectively $p_v$), and otherwise closed, independently of all other edges. Let $x=(x_1,x_2) in mathbb{Z}^2$ with $0<x_1<x_2$, and $x=(x_2,x_1)in mathbb{Z}^2$. It is natural to ask how the two point connectivity function $prob({0leftrightarrow x})$ behaves, and whether anisotropy in percolation probabilities implies the strict inequality $prob({0leftrightarrow x})>prob({0leftrightarrow x})$. In this note we give an affirmative answer in the highly supercritical regime.