Open Access
2022 On coprime percolation, the visibility graphon, and the local limit of the GCD profile
Sébastien Martineau
Author Affiliations +
Electron. Commun. Probab. 27: 1-14 (2022). DOI: 10.1214/21-ECP381


Colour an element of Zd white if its coordinates are coprime and black otherwise. What does this colouring look like when seen from a “uniformly chosen” random point of Zd? More generally, label every element of Zd by its greatest common divisor: what do the labels look like around a “uniform” random point of Zd? We answer these questions and generalisations of them, which includes a result a “local/graphon” convergence. We also investigate the percolative properties of the colouring under study.

Funding Statement

I acknowledge the support of the ERC Advanced Grant 740943 GeoBrown.


At the end of a talk given by Nathanaël Enriquez, Bálint Virág asked him about the local limit of visible lattice points: I would like to thank Bálint Virág for asking this question and Nathanaël Enriquez for letting me know of it. I am also grateful to Nathanaël Enriquez for many enthusiastic discussions about this project. I am thankful to Subhajit Goswami and Aran Raoufi for comments on an earlier version of this paper, to Samuel Le Fourn for bibliographical help, and to the anonymous referees for their feedback. I am grateful to my former postdoctoral advisors Nicolas Curien and Jean-François Le Gall, as well as the Université Paris-Sud, for having provided me with an excellent working environment.


Download Citation

Sébastien Martineau. "On coprime percolation, the visibility graphon, and the local limit of the GCD profile." Electron. Commun. Probab. 27 1 - 14, 2022.


Received: 5 February 2019; Accepted: 5 February 2021; Published: 2022
First available in Project Euclid: 28 January 2022

arXiv: 1804.06486
MathSciNet: MR4368695
Digital Object Identifier: 10.1214/21-ECP381

Primary: 05C99 , 11A05 , 20E18 , 60B10 , 60K35

Keywords: arithmetic , graphon , Local limits , percolation , visible/primitive/coprime points

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