Visible lattice points in higher dimensional random walks and biases among them

TL;DR

This paper investigates the proportion of visible lattice points in higher-dimensional random walks with specific biases, revealing that the proportion almost surely equals 1/ζ(k) and uncovering uneven distribution phenomena.

Contribution

It establishes the almost sure proportion of visible lattice points in biased higher-dimensional random walks as 1/ζ(k) and explores the distribution of visible steps, introducing new phenomena.

Findings

Proportion of visible lattice points is almost surely 1/ζ(k).

Visible steps are not evenly distributed in the walk.

Analysis combines probability theory and analytic number theory.

Abstract

For any integers $k\geq 2$, $q\geq 1$ and any finite set $\mathcal{A}=\{{\boldsymbol{\alpha}}_1,\cdots,{\boldsymbol{\alpha}}_q\}$, where ${ \boldsymbol{\alpha}_t}=(\alpha_{t,1},\cdots,\alpha_{t,k})~(1\leq t\leq q)$ with $0<\alpha_{t,1},\cdots,\alpha_{t,k}<1$ and $\alpha_{t,1}+\cdots+\alpha_{t,k}=1$, this paper concerns the visibility of lattice points in the type-$\mathcal{A}$ random walk on the lattice $\mathbb{Z}^k$. We show that the proportion of visible lattice points on a random path of the walk is almost surely $1/\zeta(k)$, where $\zeta(s)$ is the Riemann zeta-function, and we also consider consecutive visibility of lattice points in the type-$\mathcal{A}$ random walk and give the proportion of the corresponding visible steps. Moreover, we find a new phenomenon that visible steps in both of the above cases are not evenly distributed. Our proof relies on tools from probability theory and analytic number theory.