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

Abstract

For any integers $k\ge 2$, $q\ge 1$ and any finite set $\mathcal{A}=\{{\mathbf{\alpha }}_1,\cdots ,{\mathbf{\alpha }}_q\}$, where ${\mathbf{\alpha }}_t=({\alpha }_{t,1},\cdots ,{\alpha }_{t,k})(1\le t\le q)$ 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.

26 pages