Percolation of Arbitrary words on the Close-Packed Graph of $\mathbb{Z}^2$

Abstract

Let ${\Bbb Z}^2_{cp}$ be the close-packed graph of $\Bbb Z^2$, that is, the graph obtained by adding to each face of $\Bbb Z^2$ its diagonal edges. We consider site percolation on $\Bbb Z^2_{cp}$, namely, for each $v$ we choose $X(v) = 1$ or 0 with probability $p$ or $1-p$, respectively, independently for all vertices $v$ of $\Bbb Z^2_{cp}$. We say that a word $(\xi_1, \xi_2,\dots) \in \{0,1\}^{\Bbb N}$ is seen in the percolation configuration if there exists a selfavoiding path $(v_1, v_2, \dots)$ on $\Bbb Z^2_{cp}$ with $X(v_i) = \xi_i, i \ge 1$. $p_c(\Bbb Z^2, \text{site})$ denotes the critical probability for site-percolation on $\Bbb Z^2$. We prove that for each fixed $p \in \big (1- p_c(\Bbb Z^2, \text{site}), p_c(\Bbb Z^2, \text{site})\big )$, with probability 1 all words are seen. We also show that for some constants $C_i \gt 0$ there is a probability of at least $C_1$ that all words of length $C_0n^2$ are seen along a path which starts at a neighbor of the origin and is contained in the square $[-n,n]^2$.