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Source. Proposition 2.4, 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

Følner sequences, $\mathsf{cop}$, $\mu_{F,\mathsf{cop}}$ and $\mu_{\infty,\mathsf{cop}}$ are as on Proposition 2.3. For probability measures $\mu,\nu$ on $\Omega_{\{0,1\}}$, $\mu$ is stochastically dominated by $\nu$ if some coupling $(\mathcal{W},\mathcal{W}')$ of $(\mu,\nu)$ has $\mathcal{W}\subset\mathcal{W}'$ almost surely (p. 5), configurations being read as their sets of white points, as in the proof of Proposition 2.3 (p. 7), where the coupling satisfies $\omega(x)\le\omega_\infty(x)$.

Statement

Proposition 2.4 (p. 5). Let $d\ge1$ and let $(F_n)$ be a Følner sequence of $\mathbb{Z}^d$. If $\mu_{F_n,\mathsf{cop}}$ converges to some probability measure $\mu$, then $\mu$ is stochastically dominated by $\mu_{\infty,\mathsf{cop}}$.

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

The proof (pp. 6-7) records, for each prime $p$, whether a point lies outside $p\mathbb{Z}^d$; along any Følner sequence this prime-by-prime record converges to its limit law (Lemma 2.6, p. 6, deduced from Lemma 2.8, p. 8). Coprimality is the minimum over primes of these indicators, a map that is only upper semicontinuous, which yields the one-sided comparison rather than convergence (p. 6).

Dependencies

Lemma 2.6 (p. 6); Lemma 2.8 (p. 8).

Bears on

None directly; the result is an ingredient of Proposition 2.3.