A note on the phase transition for independent alignment percolation

TL;DR

This paper investigates the phase transition in an independent alignment percolation model on integer lattices, proving the critical probability for segment openness is always less than one and that the critical curve is continuous at full occupancy.

Contribution

It establishes that the phase transition is non-trivial for all occupancy probabilities and confirms the continuity of the critical curve at full occupancy, answering prior open questions.

Findings

Critical value for segment openness is less than 1 for all p in (0,1].

Phase transition is non-trivial across the entire interval (0,1].

Critical curve p → λ_c(p) is continuous at p=1.

Abstract

We study the independent alignment percolation model on $\mathbb{Z}^d$ introduced by Beaton, Grimmett and Holmes [arXiv:1908.07203]. It is a model for random intersecting line segments defined as follows. First the sites of $\mathbb{Z}^d$ are independently declared occupied with probability $p$ and vacant otherwise. Conditional on the configuration of occupied vertices, consider the set of all line segments that are parallel to the coordinate axis, whose extremes are occupied vertices and that do not traverse any other occupied vertex. Declare independently the segments on this set open with probability $\lambda$ and closed otherwise. All the edges that lie on open segments are also declared open giving rise to a bond percolation model in $\mathbb{Z}^d$. We show that for any $d \geq 2$ and $p \in (0,1]$ the critical value for $\lambda$ satisfies $\lambda_c(p)<1$ completing the proof that the phase transition is non-trivial over the whole interval $(0,1]$. We also show that the critical curve $p \mapsto \lambda_c(p)$ is continuous at $p=1$, answering a question posed by the authors in [arXiv:1908.07203].