A Note on Anisotropic Percolation

Abstract

We consider an anisotropic independent bond percolation model on $$\mathbb{Z}^{2}_{+}$$ , i.e. we suppose that the vertical edges of $$\mathbb{Z}^{2}_{+}$$ are open with probability p and closed with probability 1−p, while the horizontal edges of $$\mathbb{Z}^{2}_{+}$$ are open with probability α p and closed with probability 1−α p, with 0 < p, α < 1. Let $$x = (x_{1},x_{2})\in \mathbb{Z}^{2}_{+}$$ , with x1 < x2, and $$x^{\prime} = (x_{2}, x_{1}) \in \mathbb{Z}^{2}_{+}$$ . It is natural to ask how the two point connectivity function Pp,α({0↔ x}) behaves, and whether anisotropy in percolation probabilities implies the strict inequality Pp,α({0↔ x})> Pp,α({0↔ x′}). In this note we give affirmative answer at least for some regions of the parameters involved.