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Statement

Setting as on the Theorem 3.2 page: $C_\infty$ is the unique infinite component of the coprime pairs in $\mathbf Z^2$ under distance-1 adjacency, and $B(R)=\{z\in\mathbf Z^2:\max(|m|,|n|)<R\}$.

Theorem 3.4 (p. 51, quoted). "Let $f(R)$ be any function increasing to infinity then, except for a set of zero asymptotic density, every $(m,n)\in B(R)$ is surrounded by a rectangle of perimeter $<f(R)$ all of whose edges are contained in $C_\infty$."

The paper reads this as de Gennes' picture of the infinite component as a mesh with small holes, and suggests (p. 51), without proof, that the perimeter of the smallest such rectangle around $(m,n)$ should have a limiting distribution.

Proof pointer

Section 7, pp. 58--63; the deduction is on p. 63. Lemma 7.2 (p. 58) and Lemma 7.3 (p. 59) give the first two stages, rectangles of perimeter $O((\log R)^7)$ outside $O(R^2/\log R)$ pairs and of perimeter $O((\log\log R)^{36})$ outside $O(R^2/(\log\log R)^3)$ pairs. Lemma 7.5 (p. 61) iterates this through the iterated logarithms for $R>\exp_{k+2}(10^{15})$, each stage's rectangle edges being extended segments of coprime pairs that meet the previous stage's rectangles, which lie in $C_\infty$; the start uses the prime columns and corridors of Lemma 7.1. The segments come from Friedlander's almost-everywhere sieve (Theorems 5.1 and 5.2, p. 55), extended to intervals of general short length in the paper's Proposition 5.1 (p. 56), with Lemma 7.4 (p. 61) controlling $\sum_{p\mid m}1/p$. The proof of Theorem 3.4 chooses the number of iterations from $\log_*R-\log_*f(R)-2$ and argues by contradiction.

Read depth

Claims checked: the statement was read clause by clause on p. 51 of the edition named on the source card, and the outline of Section 7 on pp. 58--63. The proof was not checked. Nothing here is independently reviewed.

Dependencies

• Lemma 7.1 (p. 58).
• Lemmas 7.2--7.5 (pp. 58--61).
• J. B. Friedlander, Sifting short intervals, Math. Proc. Camb. Phil. Soc. 91 (1982), 9--15, the paper's reference [20], through Theorems 5.1--5.2 (p. 55) and Proposition 5.1 (p. 56).
• N. Watt, Short intervals almost all containing primes, Acta Arith. 72 (1995), 131--167, the paper's reference [48], used in Lemma 7.2.

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 describes 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.