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Source. Proposition 3.6, pp. 16-17, 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

Visibility, Følner sequences and the space $\mathfrak{X}_0$ are as on Proposition 3.5. For $y\in\mathbb{Z}^d$, $\overline{y}$ denotes its image in $\mathfrak{X}_0$, read prime by prime modulo $p$; the print uses this notation without defining it, and this is the evident reading.

Statement

Proposition 3.6 (pp. 16-17). 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 $(X_m)$ be independent uniform elements of $\mathfrak{X}_0$. Fix $M\ge1$ and $R\ge1$, and for each $n$ let $Y^n_1,\dots,Y^n_M$ be independent uniform elements of $F_n$. On pairs of indices $((m_0,y_0),(m_1,y_1))$ with $m_i\in\{1,\dots,M\}$ and $y_i\in\{-R,\dots,R\}^d$, let $\psi_n$ be the indicator that $Y^n_{m_0}+y_0$ is visible from $Y^n_{m_1}+y_1$, and let $\psi_\infty$ be the indicator that $X_{m_0}(p)+\overline{y_0}\ne X_{m_1}(p)+\overline{y_1}$ for every prime $p$. Then the law of $\psi_n$ converges to the law of $\psi_\infty$.

The proposition combines Proposition 2.3 (one point, its neighbourhood) and Proposition 3.5 (many points, their mutual visibility); this is the "local/graphon" convergence of the abstract. The paper says (p. 17) that it adapts to the whole GCD profile under a tightness assumption and to affine subspaces, and Proposition 3.7 (p. 17) extends it to profinitely closed sets in place of the coprime set, such as points whose GCD is $k$-free.

Read depth. Claims checked: the statement was read clause by clause on the print. The paper gives no separate proof.

Proof pointer

No proof is printed; the paper presents the result after stating that the arguments of Section 2.1 prove Proposition 3.5 (p. 16).

Dependencies

Proposition 2.3 and Proposition 3.5.

Bears on

None directly.