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Statement

Conjecture 1.3 (p. 277, quoted). "Consider walks along the Gaussian primes with step size at most $k\sqrt{\log|z|}$ at the prime $z=a+bi$. For any $k<\sqrt{2\pi\lambda_c}$, there is no unbounded walk, and for any $k>\sqrt{2\pi\lambda_c}$, there is an unbounded walk."

Here $\lambda_c$ is the critical intensity of the Poisson blob model, as on the Theorem 1.1 page. The paper notes (p. 277) that the existence of infinite walks as in the conjecture implies $p_{n+1}-p_n=O(\sqrt{p_n\log p_n})$ for consecutive rational primes, better than the $O(\sqrt{p_n}\log^2p_n)$ known under the Riemann Hypothesis, and that current methods are very far from such questions.

Source. Ilan Vardi, Prime percolation, Experimental Mathematics 7 (1998), no. 3, 275--289, doi:10.1080/10586458.1998.10504373: p. 277. The edition read is identified on the source card.

Read depth. Claims checked: the statement and the remark after it were read on the printed page.

Bears on

• #952: the first half of the conjecture, for any one $k$ below $\sqrt{2\pi\lambda_c}$, would imply the negative answer to the problem (an observation of this page): an infinite sequence of distinct Gaussian primes with steps at most $C$ has a tail in which every point $z$ satisfies $C\le k\sqrt{\log|z|}$, and that tail is an unbounded walk of the conjecture's kind. The paper proves no part of the conjecture.