Truncated Connectivities in a Highly Supercritical Anisotropic Percolation Model

Abstract

We consider an anisotropic bond percolation model on Z 2 , with p = (p h , pv) ∈ [0, 1] 2 , pv > p h , and declare each horizontal (respectively vertical) edge of Z 2 to be open with probability p h (respectively pv), and otherwise closed, independently of all other edges. Let x = (x1, x2) ∈ Z 2 with 0 < x1 < x2, and x ′ = (x2, x1) ∈ Z 2 . It is natural to ask how the two point connectivity function Pp({0 ↔ x}) behaves, and whether anisotropy in percolation probabilities implies the strict inequality Pp({0 ↔ x}) > Pp({0 ↔ x ′ }). In this note we give an affirmative answer in the highly supercritical regime.

A Remark on Monotonicity in Bernoulli Bond Percolation

“…The first one, about Ornstein-Zernike behaviour, was given in [8] and it is stated below as Lemma 5. The second one, on bounds upper and lower bonds for truncated two-point function was obtained in [9], is stated below as Lemma 6.…”

“…[Proposition 2 of [9]] For Z 2 , there exists p 6 < 1, close enough to 1. such that, for all p ∈ [p 6 , 1), it holds that λ 2n+2 p 2n τ f p (n) λ 2n+2 (4 3 λ) n/2+1 1 − 4 3 λ + (1 + 12λ) n .…”