Truncated Connectivities in a highly supercritical anisotropic percolation model

Abstract

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 $0 \prob(\{0\leftrightarrow x'\})$. In this note we give an affirmative answer in the highly supercritical regime.