On a property of random-oriented percolation in a quadrant

TL;DR

This paper investigates properties of random-oriented percolation models in a quadrant, establishing connections between different models and providing bounds for critical values, with extensions to higher dimensions.

Contribution

It demonstrates that various random-oriented percolation models are special cases of a broader model and provides bounds for critical values, including in higher dimensions.

Findings

Both models are special cases of NE-quadrant percolation with critical value 1/2.

Unconditional lower bound for Hegarty's model critical value.

Percolation occurs in slabs of height at least 3 in D.

Abstract

Grimmett's random-orientation percolation is formulated as follows. The square lattice is used to generate an oriented graph such that each edge is oriented rightwards (resp. upwards) with probability $p$ and leftwards (resp. downwards) otherwise. We consider a variation of Grimmett's model proposed by Hegarty, in which edges are oriented away from the origin with probability $p$, and towards it with probability $1-p$, which implies rotational instead of translational symmetry. We show that both models could be considered as special cases of random-oriented percolation in the NE-quadrant, provided that the critical value for the latter is 1/2. As a corollary, we unconditionally obtain a non-trivial lower bound for the critical value of Hegarty's random-orientation model. The second part of the paper is devoted to higher dimensions and we show that the Grimmett model percolates in any slab of height at least 3 in $\mathbb{Z}^3$.