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Source. Unnumbered paragraph closing Section 2.3 ("Remarks", pp. 11-13), p. 13, of Sébastien Martineau, "On coprime percolation, the visibility graphon, and the local limit of the GCD profile," Electronic Communications in Probability 27 (2022), 1-14, doi:10.1214/21-ECP381; arXiv:1804.06486. Pages are those of the arXiv v2 PDF named on the source card.

Setting

$\mathbb{Z}^d$ carries its usual nearest-neighbour (hypercubic) graph structure, and clusters are the connected components of the white, or of the black, points. The colouring is the limit $\mu_{\infty,\mathsf{cop}}$ of Theorem 2.1: for each prime $p$ independently one uniformly chosen coset of $p\mathbb{Z}^d$ is black, and points in no chosen coset are white. The paper states that $\mu_{\infty,\mathsf{cop}}$ is ergodic under translations and sketches why (p. 13), so the numbers of infinite white and black clusters are almost surely constant.

Statement

Remark (p. 13). For a $\mu_{\infty,\mathsf{cop}}$-random colouring:

1. for $d=2$, and hence for every $d\ge2$, there is almost surely at least one infinite white cluster; the paper derives this from Theorem 3.3 of Vardi's "Deterministic percolation" (Comm. Math. Phys. 207 (1999), 43-66);
2. for $d=2$, there is almost surely at most one infinite white cluster and no infinite black cluster; the paper derives this from Theorem 3.4 of the same paper of Vardi.

The paper states these as consequences that "One can derive" from Vardi's theorems (p. 13) and gives no derivation; they concern the random limit colouring, not the deterministic coprime set of $\mathbb{Z}^d$.

Read depth. Claims checked: the paragraph was read clause by clause on the print. Vardi's theorems were not read here, and no derivation is printed.

Proof pointer

None printed beyond the attribution to Vardi's Theorems 3.3 and 3.4 and the ergodicity argument of p. 13.

Dependencies

Theorem 2.1 (the limit colouring); Vardi's Theorems 3.3 and 3.4 (1999).

Bears on

• Problem 1212: background only. The paper does not mention the problem. Its clusters use the same adjacency as the problem's graph (points differing by one in one coordinate), but they are clusters of the random limit colouring of all of $\mathbb{Z}^2$, not of the coprime points of $\mathbb{N}^2$, and the remark does not address the problem's condition that every point on the path have a composite coordinate and both coordinates above 1.