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Statement

Setting (pp. 283--284). Section 6 considers walks along the Gaussian integers relatively prime to a fixed integer $N$, viewed modulo $N$ in the square $[0,N-1]\times[0,N-1]$, and assumes $N$ even; the setting is stated in full on the Proposition 6.1 page. The proposition names no step size; the surrounding text applies it to walks of a fixed step $k$.

Proposition 6.3 (p. 284, quoted). "There is at most one infinite connected component of Gaussian integers relatively prime to $N$."

Source. Ilan Vardi, Prime percolation, Experimental Mathematics 7 (1998), no. 3, 275--289, doi:10.1080/10586458.1998.10504373: Section 6, pp. 283--284. The edition read is identified on the source card.

Read depth. Claims checked: the setting and the statement were read clause by clause on the printed pages. The paper gives no proof.

Proof pointer

None in the paper beyond the remark (p. 284) that the fundamental square tiles the plane under the translations by $(N,0)$ and $(0,N)$, after which Propositions 6.1--6.3 are called clear.

Dependencies

None.

Bears on

None directly. The proposition concerns the Gaussian integers coprime to $N$, not the Gaussian primes, and says nothing on whether an infinite component exists.