Percolation of Arbitrary Words in $\{0, 1\}^\mathbb{N}$

Abstract

Let $\mathscr{G}$ be a (possibly directed) locally finite graph with countably infinite vertex set $\mathfrak{V}$. Let $\{X(v): v \in \mathfrak{V}\}$ be an i.i.d. family of random variables with $P\{X(v) = 1\} = 1 - P\{X(v) = 0\} = p$. Finally, let $\xi = (\xi_1, \xi_2,\ldots)$ be a generic element of $\{0, 1\}^{\mathbb{N}}$; such a $\xi$ is called a word. We say that the word $\xi$ is seen from the vertex $v$ if there exists a self-avoiding path $(v, v_1, v_2, \ldots)$ on $\mathscr{G}$ starting at $v$ and such that $X(v_i) = \xi_i$ for $i \geq 1$. The traditional problem in (site) percolation is whether $P\{(1, 1, 1, \ldots)$ is seen from $v\} > 0$. So-called $AB$-percolation occurs if $P\{(1, 0, 1, 0, 1, 0,\ldots)$ is seen from $v\} > 0$. Here we investigate (a) whether $P\{$all words are seen from $v\} = 1$. We show that both answers are positive if $\mathscr{G} = \mathbb{Z}^d$, or even $\mathbb{Z}^d_+$ with all edges oriented in the "positive direction," when $d$ is sufficiently large. We show that on the oriented $\mathbb{Z}^3_+$ the answer to (a) is negative, but we do not know the answer to (b) on $\mathbb{Z}^3_+$. Various graphs $\mathscr{G}$ are constructed (almost all of them trees) for which the set of words $\xi$ which can be seen from a given $v$ (or from some $v$) is large, even though it is w.p.1 not the set of all words.