Truncated Connectivities in a highly supercritical anisotropic percolation model

TL;DR

This paper investigates anisotropic bond percolation on a 2D lattice, demonstrating that in the highly supercritical phase, the probability of connecting two points depends on their order, confirming anisotropic effects.

Contribution

It proves that in a highly supercritical anisotropic percolation model, the two-point connectivity function exhibits strict inequalities based on point ordering.

Findings

Connectivity probability depends on point order in supercritical regime

Anisotropy causes asymmetry in connection probabilities

Results confirm anisotropic effects in highly supercritical percolation

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 $0<x_1<x_2$, and $x'=(x_2,x_1)\in \mathbb{Z}^2$. 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.