May 2022 A note on the phase transition for independent alignment percolation
Marcelo Hilário, Daniel Ungaretti
Author Affiliations +
Bernoulli 28(2): 1432-1447 (May 2022). DOI: 10.3150/21-BEJ1395


We study the independent alignment percolation model on Zd introduced by Beaton, Grimmett and Holmes. 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 d2 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 pλc(p) is continuous at p=1.

Funding Statement

The research of MH was partially supported by CNPq grants ‘Projeto Universal’ (406659/2016-8) and ‘Produtividade em Pesquisa’ (307880/2017-6) and by FAPEMIG grant ‘Projeto Universal’ (APQ-02971-17). The research of DU was partially supported by grant 2020/05555-4, São Paulo Research Foundation (FAPESP).


We would like to thank an anonymous referee for valuable suggestions that led to several simplifications in our arguments.


Download Citation

Marcelo Hilário. Daniel Ungaretti. "A note on the phase transition for independent alignment percolation." Bernoulli 28 (2) 1432 - 1447, May 2022.


Received: 1 November 2020; Revised: 1 April 2021; Published: May 2022
First available in Project Euclid: 3 March 2022

MathSciNet: MR4388944
zbMATH: 1502.60154
Digital Object Identifier: 10.3150/21-BEJ1395

Keywords: percolation , phase transition , renormalization

Rights: Copyright © 2022 ISI/BS


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Vol.28 • No. 2 • May 2022
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