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arXiv · 1512.04722

Visible lattice points in random walks

Javier Cilleruelo·José L. Fernández·Pablo Fernández
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Abstract

We consider the possible visits to visible points of a random walker moving up and right in the integer lattice (with probability $\alpha$ and $1-\alpha$, respectively) and starting from the origin. We show that, almost surely, the asymptotic proportion of strings of $k$ consecutive visible lattice points visited by such an $\alpha$-random walk is a certain constant $c_k(\alpha)$, which is actually an (explicitly calculable) polynomial in $\alpha$ of degree $2\lfloor(k-1)/2\rfloor $. For $k=1$, this gives that, almost surely, the asymptotic proportion of time the random walker is visible from the origin is $c_1(\alpha)=6/\pi^2$, independently of $\alpha$.

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Javier Cilleruelo, José L. Fernández, Pablo Fernández. 2015-12-15. Visible lattice points in random walks. https://arxiv.org/abs/1512.04722

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