Vardi 1999 deterministic percolation

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lemma_7_1: Vardi's lemma that the infinite component of the coprime lattice points contains the points (m,1) with m > 0, the points (p,n) with p prime and p > n, and the points (m,q) with q prime, q < m < (q/2)^(20/11) and q not dividing m.

proposition_3_1: Vardi's elementary proposition that the set R of coprime integer pairs in the whole plane, two sites joined when at Euclidean distance 1, has exactly one infinite component.

theorem_3_2: Vardi's main theorem that the infinite component of the set of coprime integer pairs under distance-1 adjacency has an asymptotic density, taken over the squares max(|m|,|n|) < R.

theorem_3_3: Vardi's theorem that the asymptotic density of the infinite component of the coprime integer pairs under distance-1 adjacency, which exists by his Theorem 3.2, is positive.

theorem_3_4: Vardi's theorem that for any function f(R) increasing to infinity, every point of the square B(R) outside a set of zero asymptotic density is surrounded by a rectangle of perimeter less than f(R) whose edges lie in the infinite component of the coprime lattice points.

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Ilan Vardi, Deterministic percolation. Communications in Mathematical Physics 207 (1999), 43-66, DOI 10.1007/s002200050717. The copy read for this card is an author-hosted copy of the publisher's version, which prints "© Springer-Verlag 1999" on its first page, every other right reserved.

Vardi poses percolation questions for the deterministic set R = {(m,n) in Z^2 : gcd(m,n) = 1}, sites joined when at Euclidean distance 1, and records (p. 44) that the connectivity of R was posed as a problem in Erdős, Gruber and Hammer, Lattice Points (1989), p. 109. Densities are taken over the squares B(R) = {max(|m|,|n|) < R} (pp. 47-48). Proposition 3.1 (p. 50) proves elementarily that R has a unique infinite component C_infinity: the line {(m,1) : m >= 1} meets every prime column {(p,n) : 1 <= n <= p-1}, so any infinite component in the region {m > n} crosses one of them, and the eight lines {(+-1,+-k)}, {(+-k,+-1)} are joined at (+-1,+-1) or through (+-1,0) and (0,+-1), with gcd(1,0) = 1. The main results are that C_infinity has an asymptotic density (Theorem 3.2, p. 51, proved in Section 8, pp. 64-65) and that this density is not zero (Theorem 3.3, p. 51, proved on p. 63); preliminary computations reported on p. 51 suggest about 96% of open sites lie on it, and the same page proves the upper bound (1 - 1/144) 6/pi^2. Both rest on Theorem 3.4 (p. 51): for any f increasing to infinity, every point of B(R) outside a set of zero asymptotic density is surrounded by a rectangle of perimeter less than f(R) whose edges lie in C_infinity, matching de Gennes' picture of a mesh with small holes. Theorem 3.2 needs only the weaker Lemma 7.3 (p. 59): all but O(R^2/(log log R)^3) pairs of B(R) are surrounded by such a rectangle of perimeter O((log log R)^36). Lemma 7.1 (p. 58) supplies the starting sets: the line {(m,1) : m > 0}, the prime columns {(p,n) : p > n}, and, through the Heath-Brown–Iwaniec theorem on primes in intervals of length y^{11/20}, the horizontal corridors {(m,q) : q < m < (q/2)^{20/11}, q not dividing m} at prime heights q. The engine is an 'almost everywhere' sieve of Friedlander (Theorems 5.1-5.2, p. 55), extended to intervals of general short length in the paper's Proposition 5.1 (p. 56), together with Watt's result that almost every interval of length y^{1/14+epsilon} contains a prime. The paper's bibliography does not include Erdős's 1980 survey or Herzog and Stewart's 1971 paper.

Source: https://www.lix.polytechnique.fr/Labo/Ilan.Vardi/.

Bears on. #1212: the paper studies the infinite component of the coprime pairs under the problem's adjacency, taken over all of Z^2 rather than N^2, with no restriction on the coordinates. The component it builds runs along the lines with a coordinate +-1 and along lines with a prime coordinate (Proposition 3.1, Lemma 7.1). The paper does not consider paths that avoid coordinate 1 or pairs of primes, so it does not address the problem's question.

Results.

• Proposition 3.1 (p. 50): R has a unique infinite component.
• Theorem 3.2 (p. 51): the infinite component of R has an asymptotic density.
• Theorem 3.3 (p. 51): that asymptotic density is not zero.
• Theorem 3.4 (p. 51): for any f(R) increasing to infinity, all points of B(R) outside a set of zero asymptotic density are surrounded by a rectangle of perimeter less than f(R) whose edges lie in the infinite component.
• Lemma 7.1 (p. 58): the line at height 1, the prime columns and the corridors at prime heights q with q < m < (q/2)^{20/11}, q not dividing m, lie in the infinite component.

Read status: claims checked for the five results above, statements read clause by clause against the print; proofs of Proposition 3.1 and Lemma 7.1 read, the others not checked.

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