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Vardi, Ilan

Prime percolation. (English) Zbl 0926.11065

Exp. Math. 7, No. 3, 275-289 (1998).
Is there an unbounded walk along Gaussian primes of step size \(\leq L\) ? Not for \(L\leq \sqrt{26}\), and probably not for any bounded \(L\), due to their local density.
To investigate this further, the author constructs here a random model that mimicks the Gaussian primes. He then employs percolation theory to show that an unbounded walk of local step size \(L= k\sqrt{\log| z|}\) at a point \(z\) exists (does not exist) if \(k> \sqrt{2\pi \lambda_c}\) (if \(k< \sqrt{2\pi \lambda_c}\), both with probability 1. Here, \(\lambda_c \approx 0.35\) is a constant from continuum percolation of a Poisson process, see Ch. 12.10 of G. Grimmett, Percolation, 2nd ed., Springer, Berlin (1999) for background material.
Based on this result and other known facts in this area, the author conjectures that the critical step size for the actual Gaussian primes is also \(\sqrt{2\pi \lambda_c \log| z|}\) (at \(z\)).
Reviewer: M.Baake (Tübingen)

Cited in 1 Review
Cited in 3 Documents

MSC:

11N05 Distribution of primes
82B43 Percolation
11K99 Probabilistic theory: distribution modulo \(1\); metric theory of algorithms
60K35 Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

unbounded walk along Gaussian primes; percolation; critical step size

Citations:

Zbl 0990.28733
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Full Text: DOI EuDML EMIS

References:

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