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Statement

Setting (pp. 276--277). The random model declares each Gaussian integer $z$ with $|z|>2$ open, independently, with probability $2/(\pi\log|z|)$: for distinct Gaussian integers $z_1,\ldots,z_n,z'_1,\ldots,z'_m$ of modulus above $2$, the probability that all $z_j$ are open and all $z'_k$ closed is

$$\prod_{j=1}^{n}\frac{2}{\pi\log|z_j|}\ \prod_{k=1}^{m} \Bigl(1-\frac{2}{\pi\log|z'_k|}\Bigr).$$

Section 4 (p. 281) restates the model with $|z|>1$ in place of $|z|>2$. The constant $\lambda_c$ (p. 276) is the critical intensity of the Poisson blob model of continuum percolation: a Poisson process of intensity $\lambda$ in the plane, a disk of radius one about each point, and an unbounded connected set of disks with probability one for $\lambda>\lambda_c$ and probability zero for $\lambda<\lambda_c$ (Zuev and Sidorenko, 1985). The paper reports that $\lambda_c$ is believed to be about $0.35$ (p. 276) and that the best proved bounds are $0.174<\lambda_c<0.843$ (Hall, 1985; p. 281).

Theorem 1.1 (p. 277, quoted). "Consider the Gaussian integers with the above probability model and consider walks of step size at most $k\sqrt{\log|z|}$ at $z$, where $k$ is a constant. Then for $k<\sqrt{2\pi\lambda_c}$, with probability one, there is no unbounded open component, and for $k>\sqrt{2\pi\lambda_c}$, with probability one, there is an unbounded open component."

The step bound grows with the modulus of the point; the theorem says nothing at $k=\sqrt{2\pi\lambda_c}$, and it is a statement about the random model, not about the Gaussian primes. Its transfer to the Gaussian primes is Conjecture 1.3.

Source. Ilan Vardi, Prime percolation, Experimental Mathematics 7 (1998), no. 3, 275--289, doi:10.1080/10586458.1998.10504373: the model and Theorem 1.1 on pp. 276--277, the proof in Section 5, pp. 282--283. The edition read is identified on the source card.

Read depth. Claims checked: the model and the statement were read clause by clause on the printed pages. The proof was read for its structure only; no step was checked. Nothing here is independently reviewed.

Proof pointer

Section 5 (pp. 282--283). The map $f_s(z)=z/(s\sqrt{\log|z|})$ carries the open points of the model approximately to a Poisson process of intensity $\lambda=2s^2/\pi$, and a disk of radius $s\sqrt{\log|z|}$ about $z$ to a disk of radius about one, so the critical value is $s_c=\sqrt{\pi\lambda_c/2}$; a walk of step $k\sqrt{\log|z|}$ corresponds to overlapping disks of radius $k/2$, giving $k=2s_c=\sqrt{2\pi\lambda_c}$. Part (a) treats $s>\sqrt{\pi\lambda_c/2}$ by carrying an infinite component of a Poisson blob model of intensity $\lambda_1$, with $\lambda>\lambda_1>\lambda_c$, back to the Gaussian integers; part (b) treats $s<\sqrt{\pi\lambda_c/2}$ by contradiction, comparing with the subcritical blob model of intensity $2s_1^2/\pi$ for some $s<s_1<\sqrt{\pi\lambda_c/2}$.

Dependencies

The continuum percolation theorem of Zuev and Sidorenko (Teoret. Mat. Fiz. 62 (1985), 76--86), cited through Grimmett's Percolation, Section 10.5.

Bears on

• #952: none as a proof. The theorem concerns a random model with step bounds growing like $\sqrt{\log|z|}$; the paper draws from it Conjecture 1.3 (p. 277), whose first half would give the negative answer to the problem, and it establishes nothing about the Gaussian primes themselves.