Alignment Percolation

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