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Statement

The paper's setting (p. 44): $\mathcal R=\{(m,n)\in\mathbf Z^2:\gcd(m,n)=1\}$, two sites joined when they are at Euclidean distance $1$, that is, when they differ by $1$ in exactly one coordinate. The proof (p. 50) uses the convention $\gcd(1,0)=1$, so $(\pm1,0)$ and $(0,\pm1)$ lie in $\mathcal R$.

Proposition 3.1 (p. 50, quoted). "$\mathcal R$ has a unique infinite component."

Existence alone is noted as trivial on the same page, since the line $\{(m,1):m=1,2,3,\dots\}$ lies in $\mathcal R$. The paper later writes $C_\infty$ for this component.

Proof pointer

p. 50, elementary. Let $C_1$ be the component containing the line $\{(m,1):m\ge1\}$. For every prime $p$ the vertical segment $\{(p,n):1\le n\le p-1\}$ consists of coprime pairs and meets that line, so it lies in $C_1$. An infinite component inside the region $m>n$ must eventually cross one of these segments and so equals $C_1$. The same holds by symmetry in the other seven octant regions, around the lines $\{(\pm1,\pm k)\}$ and $\{(\pm k,\pm1)\}$, and these eight lines are joined to one another at $(\pm1,\pm1)$ or through $(\pm1,0)$ and $(0,\pm1)$.

Read depth

Claims checked: the statement, the definitions it uses and the proof were read clause by clause on p. 50 of the edition named on the source card. Nothing here is independently reviewed.

Dependencies

None beyond the definitions of Section 3.

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 proposition concerns the same adjacency on coprime pairs, taken over all of $\mathbf Z^2$ and with no restriction on the coordinates. The component it identifies is built from the line with second coordinate $1$ and from segments with a prime first coordinate. The paper does not consider paths that avoid coordinate $1$ or pairs of primes, so the proposition does not address the problem's question.