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Statement

Theorem 7.1 (p. 285, attributed to Gethner and Stark 1997, quoted). "There is no unbounded walk of step length $\sqrt2$."

The walk is along the Gaussian primes. Section 7 (p. 284) treats such walks as walks on the odd Gaussian integers, a square lattice on which step $\sqrt2$ joins nearest neighbours. The paper adds (p. 285) that the result also follows from a stronger result of Jordan and Rabung (J. Number Theory 8 (1976), 43--51), recorded as its Theorem 7.2: the largest admissible $\sqrt2$-connected component has size 48. A remark on p. 287 adds that Gethner and Stark also showed there is no walk of step size $2$, proving Conjecture 1.2 in that case.

Source. Ilan Vardi, Prime percolation, Experimental Mathematics 7 (1998), no. 3, 275--289, doi:10.1080/10586458.1998.10504373: Section 7, pp. 284--285, and the remark on p. 287. The edition read is identified on the source card.

Read depth. Claims checked: the statement and the deduction were read on the printed pages; the check shown in Figure 7 was not repeated, and the cited proofs were not read.

Proof pointer

Pages 284--285. For $N=2M=130=2\cdot5\cdot13$ the density of odd Gaussian integers coprime to $N$ is $\delta(M)=0.545325\ldots$, below the site percolation threshold $\rho_c(1)\approx0.59$ of the square lattice. Figure 7 (p. 285) draws the Gaussian integers coprime to $130$ in the fundamental triangle, which the text calls $F(65)$ although the figure spans $0\le a\le65$, the triangle $F(130)$ of the setting on p. 284. No connected set of disks of radius $1/\sqrt2$ about these points touches all three sides, so Proposition 6.1 rules out a walk to infinity.

Dependencies

Proposition 6.1 of the same paper, and the computation shown in Figure 7.

Bears on

• #952: the negative answer for step bound $\sqrt2$ only, a case contained in Gethner and Stark's step-2 result recorded on its claim page.