We study the independent alignment percolation model on introduced by Beaton, Grimmett and Holmes. It is a model for random intersecting line segments defined as follows. First the sites of 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 . We show that for any and the critical value for λ satisfies completing the proof that the phase transition is non-trivial over the whole interval . We also show that the critical curve is continuous at .
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.
"A note on the phase transition for independent alignment percolation." Bernoulli 28 (2) 1432 - 1447, May 2022. https://doi.org/10.3150/21-BEJ1395