Bernoulli line percolation

TL;DR

This paper introduces a new percolation model on high-dimensional integer lattices where entire coordinate lines are randomly removed, revealing phase transitions and decay behaviors in the resulting vacant set.

Contribution

It establishes the existence of a phase transition in the vacant set and characterizes decay properties and the number of infinite components in this novel line percolation model.

Findings

Existence of a phase transition for the vacant set.

Power-law decay of connectivity in the infinite component region.

Transition from exponential to power-law decay outside the infinite component region.

Abstract

We introduce a percolation model on $\mathbb{Z}^d$, $d \geq 3$, 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 $\mathcal{V}$ if and only if none of the $d$ lines to which it belongs, is removed. We show the existence of a phase transition for $\mathcal{V}$ as the probability of removing the lines is varied. We also establish that, in the certain region of parameters space where $\mathcal{V}$ contains an infinite component, the truncated connectivity function has power-law decay, while inside the region where $\mathcal{V}$ 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 $\mathcal{V}$ has an infinite connected component. We also show that the number of infinite connected components of $\mathcal{V}$ is either $0$, $1$ or $\infty$.