2015/09/21 by Hilário, Marcelo R., Sidoravicius, Vladas · 2 citations
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1509.06204
We introduce a percolation model on ℤd, d ≥ 3, in which the discrete lines of vertices that are parallel to the coordinate axis are entirely removed at random and independently of each other. In this way a vertex belongs to the vacant set V if and only if none of the d lines to which it belongs, is removed. We show the existence of a phase transition for V as the probability of removing the lines is varied. We also establish that, in the certain region of parameters space where V contains an infinite component, the truncated connectivity function has power-law decay, while inside the region where V has no infinite component, there is a transition from exponential to power-law decay. In the particular case d=3 the power-law decay extends through all the region where V has an infinite connected component. We also show that the number of infinite connected components of V is either 0, 1 or ∞.