Percolative properties of the random coprime colouring

2025/09/10 by Samuel Le Fourn, Fourn, Samuel Le, Mike Liu +3
Mathematics · #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2509.08452

openalex publication_date 2025/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given u and v in ℤ^d, say that u is visible from v if the segment from u to v contains exactly two elements, which are u and v. Take X "uniformly at random in ℤ^d" and colour each vertex u of ℤ^d in white if u is visible from X and in black otherwise. Previous independent works of Pleasants-Huck and of the third author give a precise meaning to this definition. This paper is dedicated to the study of this random colouring from the point of view of percolation theory: given a reasonable graph structure on ℤ^d, how many infinite black (resp. white) connected components are there?

Citations

• Stochastic domination and lifts of random variables in percolation theory
• Universal optimality of the E₈ and Leech lattices and interpolation formulas
• On coprime percolation, the visibility graphon, and the local limit of the GCD profile
• Bernoulli line percolation
• A formal proof of the Kepler conjecture
• Oriented percolation in a random environment
• A proof of the Kepler conjecture

Related

• On coprime percolation, the visibility graphon, and the local limit of the GCD profile
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• Color-avoiding percolation on the Erdős-Rényi random graph
• Random colourings of aperiodic graphs: Ergodic and spectral properties
• Merging percolation on Z^d and classical random graphs: Phase transition