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Source. Proposition 3.5, p. 16, 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

Two distinct points $x,y\in\mathbb{Z}^d$ are visible from each other when the segment $[x,y]$ meets $\mathbb{Z}^d$ only at $x$ and $y$, that is, when $\gcd(x-y)=1$ (pp. 2-3). Følner sequences are as on Proposition 2.3.

A graphon here is represented by a standard probability space and a symmetric measurable function from its square to $[0,1]$, up to measure-preserving isomorphism (p. 15). A sequence of random finite graphs $\mathcal{G}_n=(V_n,E_n)$ with $|V_n|$ tending to infinity in probability converges to the graphon represented by $(\mathfrak{X},\mathbb{P},f)$ if, for every $k$, the edge indicators among $k$ independent uniform vertices converge in law to $(f(X_i,X_j))_{1\le i<j\le k}$ with $X_1,\dots,X_k$ independent of law $\mathbb{P}$ (pp. 15-16).

The space $\mathfrak{X}_0=\prod_{p}(\mathbb{Z}/p\mathbb{Z})^d$, over all primes $p$, carries the product of uniform measures, and $\delta(x_1,x_2)=1$ when $x_1(p)\ne x_2(p)$ for every prime $p$, and $0$ otherwise (p. 16).

Statement

Proposition 3.5 (p. 16). Let $d\ge1$ and let $(F_n)$ be a Følner sequence of $\mathbb{Z}^d$ such that the probability that a uniform point of $F_n$ is coprime converges to $1/\zeta(d)$. Let $\mathcal{G}_n$ be the random graph on vertex set $F_n$ in which two distinct vertices are joined exactly when one is visible from the other. Then $\mathcal{G}_n$ converges to the graphon represented by $(\mathfrak{X}_0,\delta)$.

Read depth. Claims checked: the statement and definitions were read clause by clause on the print. The paper gives no separate proof; it says the result follows by the arguments of Section 2.1 (p. 16).

Proof pointer

The print states that the proof follows the arguments of Section 2.1 (pp. 4-7), which prove Proposition 2.3.

Dependencies

The method of Proposition 2.3 and Proposition 2.4. The combined statement is Proposition 3.6.

Bears on

None directly.