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Source. Proposition 2.7, p. 8, 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

Følner sequences are as on Proposition 2.3; $\mathsf{gcd}$, $\mu_{F,\mathsf{gcd}}$ and the limit $\mu_{\infty,\mathsf{gcd}}$ are as on Theorem 1.1.

Statement

Proposition 2.7 (p. 8). Let $d\ge1$, let $(F_n)$ be a Følner sequence of $\mathbb{Z}^d$, and let $Y_n$ be uniform in $F_n$. Assume that the laws of $\gcd(Y_n)$ form a tight sequence; the print notes that this forces $d\ge2$. Then $\mu_{F_n,\mathsf{gcd}}$ converges to $\mu_{\infty,\mathsf{gcd}}$.

The paper notes (p. 11) that the conclusion implies that of Proposition 2.3, but the hypotheses differ: Fact 2.12 (p. 11) gives Følner sequences with coprime proportion tending to $1/\zeta(d)$ and non-tight GCD.

Read depth. Claims checked: the statement was read clause by clause on the print. The proof was read but not checked step by step.

Proof pointer

Along any Følner sequence, the view of the profinite coordinates converges to a Haar-distributed translate (Lemma 2.8, p. 8), hence the GCD read as a supernatural number (an exponent in $\{0,\dots,\infty\}$ for each prime) converges too (Lemma 2.10, p. 9). Tightness gives a subsequential limit on $\Omega_{\mathbb{N}}$, and an injectivity lemma (Lemma 2.11, p. 10) identifies it with $\mu_{\infty,\mathsf{gcd}}$ (proof, p. 10).

Dependencies

Lemmas 2.8 (p. 8), 2.10 (p. 9) and 2.11 (p. 10).

Bears on

None directly; the result yields Theorem 1.1.