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Statement

Setting (pp. 44, 47--48): $\mathcal R=\{(m,n)\in\mathbf Z^2:\gcd(m,n)=1\}$ with sites joined at Euclidean distance $1$, and $C_\infty$ its unique infinite component (Proposition 3.1). Densities use square summation: $B(R)=\{z\in\mathbf Z^2:\|z\|<R\}$ with $\|(m,n)\|=\max(|m|,|n|)$, and the asymptotic density of an event $P$ is the limit as $R\to\infty$ of $|\{z\in\mathcal R\cap B(R):P(z)\}|/|B(R)|$.

Theorem 3.2 (p. 51, quoted). "The infinite component of $\mathcal R$ has an asymptotic density."

In the notation of Section 8 (p. 64), the limit $\theta=\lim_{R\to\infty}|B(R)\cap C_\infty|/|B(R)|$ exists. The paper reports (p. 51) that preliminary computations suggest $\theta/p(\mathcal R_2)\approx.96\pm.01$, about 96% of the open sites, and proves the upper bound $\theta\le6/\pi^2-4\delta(\gamma)=(1-1/144)\,6/\pi^2$, where $\gamma(z)$ holds when $z\equiv(4,15)\pmod{30}$ and such $z$ in $\mathcal R$ are isolated.

Proof pointer

Section 8, pp. 64--65. The infinite component of $\mathcal R$ reduced modulo $h$ is characterized locally (Lemma 8.1); its density $\theta_h$ is non-increasing along the primorials $h=P(X)$ (Lemma 8.2), and bounds $\theta(R)$ above up to $o(1)$ (Lemma 8.3), so $\theta_*=\lim\theta_{P(X)}$ exists. Lemma 8.4 shows $\theta(R)\to\theta_*$: passing from $\mathcal R$ modulo $P(X)$, with $X\asymp\log R$, to $\mathcal R$ in $B(R)$ removes $O(R^2/\log R)$ sites, and by Lemma 7.3 almost every site is enclosed by a rectangle in $C_\infty$ small enough that the removals disconnect a vanishing proportion. The paper notes (p. 51) that this needs only Lemma 7.3 rather than the full Theorem 3.4.

Read depth

Claims checked: the statement and definitions were read clause by clause on pp. 47--48, 51 and 64 of the edition named on the source card. The proof was read but not checked. Nothing here is independently reviewed.

Dependencies

• Lemma 7.3 (p. 59): all but $O(R^2/(\log\log R)^3)$ pairs of $B(R)$ are surrounded by a rectangle of perimeter $O((\log\log R)^{36})$ whose edges lie in $C_\infty$.
• Lemmas 8.1--8.4 (pp. 64--65).

Source. Ilan Vardi, "Deterministic Percolation," Communications in Mathematical Physics 207 (1999), 43--66, DOI 10.1007/s002200050717, the edition read for the source card.

Bears on

• Problem 1212: the theorem concerns the infinite component of the problem's graph taken over all of $\mathbf Z^2$ with no restriction on the coordinates. It does not consider paths that avoid coordinate $1$ or pairs of primes and does not address the problem's question.