Truncated Connectivities in a highly supercritical anisotropic percolation model

Abstract

We consider an anisotropic bond percolation model on ${\mathbb{Z}}^2$, with $\text{p}=(p_h,p_v)\in [0,1{]}^2$, $p_v>p_h$, 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 $0<x_1<x_2$, and $x^{\prime }=(x_2,x_1)\in {\mathbb{Z}}^2$. It is natural to ask how the two point connectivity function $\prob(\{0\leftrightarrow x\})$ behaves, and whether anisotropy in percolation probabilities implies the strict inequality $\prob(\{0\leftrightarrow x\})>\prob(\{0\leftrightarrow x'\})$. In this note we give an affirmative answer in the highly supercritical regime.

11 pages

References in corpus (1)

• Percolation on infinite graphs and isoperimetric inequalities

Cited by in corpus (1)

• A remark on monotonicity in Bernoulli bond Percolation