2020/07/01 by Marcelo R. Hilário, Hilário, Marcelo, Daniel Ungaretti +1
Mathematics · Physics and Astronomy · #60K35 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · doi:10.48550/arxiv.2007.00539
openalex publication_date 2020/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the independent alignment percolation model on ℤd 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 ℤd 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 λ and closed otherwise. All the edges that lie on open segments are also declared open giving rise to a bond percolation model in ℤd. We show that for any d ≥ 2 and p ∈ (0,1] the critical value for λ satisfies λc(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 ↦ λc(p) is continuous at p=1, answering a question posed by the authors in [arXiv:1908.07203].