Corner percolation with preferential directions

TL;DR

This paper studies a dependent bond percolation model on Z^2 with preferential directions, proving the existence of infinitely many infinite paths with a common asymptotic slope under certain biased conditions.

Contribution

It introduces a regime with biased directional probabilities in corner percolation and proves the almost sure existence of infinitely many infinite paths with a shared asymptotic slope.

Findings

Existence of infinitely many infinite paths under biased probabilities

All infinite paths share the same asymptotic slope

Paths are almost surely infinite and connected

Abstract

Corner percolation is a dependent bond percolation model on Z^2 introduced by B\'alint T\'oth, in which each vertex has exactly two incident edges, perpendicular to each other. G\'abor Pete has proven in 2008 that under the maximal entropy probability measure, all connected components are finite cycles almost surely. We consider here a regime where West and North directions are preferred with probability p and q respectively, with (p,q) different from (1/2,1/2). We prove that there exists almost surely an infinite number of infinite connected components, which are in fact infinite paths. Furthermore, they all have the same asymptotic slope (2q-1)/(1-2p).