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Statement

Setting as on the Theorem 3.2 page: $\mathcal R$ is the set of coprime pairs in $\mathbf Z^2$ with distance-1 adjacency, $C_\infty$ its unique infinite component, and densities are taken over the squares $B(R)=\{\max(|m|,|n|)<R\}$.

Theorem 3.3 (p. 51, quoted). "The asymptotic density of the infinite component of $\mathcal R$ is not zero."

No explicit lower bound is given.

Proof pointer

p. 63, assuming Theorem 3.2. If the density $\theta$ were zero there would be an $f(R)\to\infty$ with $\theta(R)<1/f(R)$ for all large $R$. Theorem 3.4, applied with $\sqrt{f(R)}$, surrounds almost every point of $B(R)$ by a rectangle of perimeter $\sqrt{f(R)}$ with edges in $C_\infty$, which gives $\theta(R)\gg1/\sqrt{f(R)}$, a contradiction. The paper also says (p. 59) that its Lemma 7.3 is included to give a self-contained proof of this theorem.

Read depth

Claims checked: the statement and the proof on p. 63 were read clause by clause in the edition named on the source card; the proof of Theorem 3.4 it rests on was not checked. Nothing here is independently reviewed.

Dependencies

• Theorem 3.2 (p. 51), for the existence of the density.
• Theorem 3.4 (p. 51).

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.