Alignment percolation

TL;DR

This paper investigates the existence of infinite clusters in two stochastic models of intersecting line segments on a lattice, analyzing phase diagrams and cluster properties in dimensions two and higher.

Contribution

It introduces and analyzes two new models of line segment percolation based on site percolation, establishing phase diagrams and cluster behaviors.

Findings

Phase diagrams characterized for both models.

Conditions for infinite cluster existence identified.

Differences between models highlighted.

Abstract

The existence (or not) of infinite clusters is explored for two stochastic models of intersecting line segments in $d \ge 2$ dimensions. Salient features of the phase diagram are established in each case. The models are based on site percolation on ${\mathbb Z}^d$ with parameter $p\in (0,1]$. For each occupied site $v$, and for each of the $2d$ possible coordinate directions, declare the entire line segment from $v$ to the next occupied site in the given direction to be either blue or not blue according to a given stochastic rule. In the one-choice model, each occupied site declares one of its $2d$ incident segments to be blue. In the independent model, the states of different line segments are independent.