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Article Dans Une Revue Electronic Communications in Probability Année : 2022

On coprime percolation, the visibility graphon, and the local limit of the GCD profile

Résumé

Colour an element of Z^d white if its coordinates are coprime and black otherwise. What does this colouring look like when seen from a "uniformly chosen" point of Z^d ? Our answer to this question opens up further fields of investigation. One can generalise this problem in several directions (look from a point uniformly sampled in a subgroup, study the N-valued colouring by the gcd, etc.): we solve the generalised problems, provide results of graphon convergence, as well as a "local/graphon" convergence. One can also study the percolative properties of the colouring under study.
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Dates et versions

hal-03987170 , version 1 (14-02-2023)

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Sébastien Martineau. On coprime percolation, the visibility graphon, and the local limit of the GCD profile. Electronic Communications in Probability, 2022, 27, ⟨10.1214/21-ECP381⟩. ⟨hal-03987170⟩
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