Visible lattice points in random walks

TL;DR

This paper studies the long-term behavior of a random walk on the integer lattice, focusing on the frequency of visits to visible points, and finds explicit formulas for these proportions depending on the walk's bias.

Contribution

It introduces explicit formulas for the asymptotic proportions of visits to visible lattice points in a biased random walk, revealing their dependence on the walk's parameters.

Findings

Asymptotic proportion of visits to visible points is a constant $c_k(eta)$ for each $k$ and bias $eta$.

For $k=1$, the proportion is $6/\pi^2$, independent of bias.

Explicit polynomial formulas for $c_k(\alpha)$ are derived for all $k$.

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 $\alpha$.