Alignment Percolation

Abstract

The existence (or not) of infinite clusters is explored for two stochastic models of intersecting line segments in d≥ 2 dimensions. Salient features of the phase diagram are established in each case. The models are based on site percolation on ℤd with parameter p ∈ (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.