Fourn 2025 percolative properties random coprime colouring

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Samuel Le Fourn, Mike Liu, Sébastien Martineau, Percolative properties of the random coprime colouring. arXiv preprint (2025). arXiv:2509.08452. The arXiv record (https://arxiv.org/abs/2509.08452, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

The paper studies percolation for the random coprime coloring of a lattice Gamma in R^d: for each prime p an independent uniform coset B_p of p*Gamma is chosen, W is the complement of the union of the B_p, and vertices in W are white (visible from a 'uniformly random' point) and the rest black. Theorem 1.1 proves that for Gamma = Z^d with the usual nearest-neighbor graph and d >= 2 there is almost surely exactly one infinite white cluster and almost surely every black cluster is finite; Remark 1.2 notes d = 1 is degenerate since W is almost surely empty. Proposition 1.5 gives existence of an infinite white cluster for every Cayley graph of a lattice in dimension at least 3, so only the planar case of Theorem 1.1 is substantial for existence. With minimal-norm generators, Theorem 1.6 proves that exactly one infinite white cluster exists almost surely for the triangular, D_d (d >= 2), E_8 and Leech lattices, and Theorem 1.7 proves that no infinite black cluster exists for the triangular and D_d lattices only; Theorem 1.8 proves that exactly one infinite white cluster exists almost surely for spread-out l^p-ball generating sets on Z^d, where an infinite black cluster can exist (it does as soon as alpha >= 2). The proof of Theorem 1.1 is direct and elementary, and must avoid the Burton-Keane argument because the coloring is not insertion-tolerant (Section 2.4). For problem 1212 this is the citation-sweep item flagged in the review note: Remark 1.3 records that Theorem 1.1 was previously derivable only indirectly, by combining Martineau's earlier main theorem with Vardi's work built on Friedlander, whereas here the infinite-component statement gets a short elementary proof.

Source: https://arxiv.org/abs/2509.08452.

Bears on. #1212

Results to transcribe.

• Theorem 1.1: For d >= 2 and Z^d with its nearest-neighbor graph, almost surely the white vertices form exactly one infinite cluster and every black cluster is finite.
• Remark 1.3: Theorem 1.1 was previously obtainable only indirectly via Martineau's main theorem plus Vardi and Friedlander; this paper gives a short direct elementary proof.
• Proposition 1.5: For d >= 3 and any admissible generating set of any lattice in R^d, an infinite white cluster exists almost surely.
• Theorem 1.6: For the triangular, D_d, E_8 and Leech lattices with minimal-norm generators, the infinite white cluster almost surely exists and is unique.
• Theorem 1.7: For the triangular and D_d lattices with minimal-norm generators, almost surely every black cluster is finite.
• Theorem 1.8: For d >= 2, p in [1, infinity], alpha in [1, infinity) and the Cayley graph on Z^d with S the nonzero points of l^p-norm at most alpha, the infinite white cluster almost surely exists and is unique.