Visible lattice points in random walks

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 $\alpha$.