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Source. Proposition 2.3, p. 5, 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

A sequence $(F_n)$ of finite nonempty subsets of $\mathbb{Z}^d$ is a Følner sequence if $|F_n\Delta(F_n+y)|=o(|F_n|)$ for every $y\in\mathbb{Z}^d$ (p. 5). The paper does not require $(F_n)$ to be monotone or to exhaust $\mathbb{Z}^d$ (p. 11). The colouring $\mathsf{cop}$, the measures $\mu_{F,\mathsf{cop}}$ and the limit $\mu_{\infty,\mathsf{cop}}$ are as on Theorem 2.1.

Statement

Proposition 2.3 (p. 5). Let $d\ge1$ and let $(F_n)$ be a Følner sequence of $\mathbb{Z}^d$. Assume that the probability that the origin is white under $\mu_{F_n,\mathsf{cop}}$, that is, the proportion of coprime points in $F_n$, converges to $1/\zeta(d)$. Then $\mu_{F_n,\mathsf{cop}}$ converges to $\mu_{\infty,\mathsf{cop}}$.

The print writes the hypothesis as "$\mu_{F_n}(\{\omega : (0,\ldots,0)\in\omega\})$ converges to $1/\zeta(d)$", omitting the subscript $\mathsf{cop}$; the reading above is the one used in the proof (p. 7).

Sharpness (p. 12). Neither hypothesis can be removed. Remark 2.13 notes that, by the Chinese Remainder Theorem, there are arbitrarily large boxes $x_n+\llbracket0,N\rrbracket^d$ with no coprime point; such boxes form a Følner sequence along which $\mu_{F_n,\mathsf{cop}}$ converges to the all-black colouring. It adds that the Følner condition cannot be removed either. Fact 2.12 (p. 11) shows, for every $d\ge1$, a Følner sequence in which the coprime proportion tends to $1/\zeta(d)$ but the GCD of a uniform point is not tight, so this proposition does not follow from Proposition 2.7.

Read depth. Claims checked: the statement, Fact 2.12 and Remark 2.13 were read clause by clause on the print. The proof was read but not checked step by step.

Proof pointer

By compactness one may pass to a subsequential limit $\mu$ (p. 7). Proposition 2.4 gives a coupling in which the $\mu$-colouring is white only where the $\mu_{\infty,\mathsf{cop}}$-colouring is white. Both measures give the origin probability $1/\zeta(d)$ of being white, and both are translation invariant, so the two colourings agree almost surely.

Dependencies

Proposition 2.4 (p. 5).

Bears on

• Problem 1212: background only. The paper does not mention the problem; the proposition concerns the law of the coprime colouring in finite windows, not paths through coprime points.