Visible lattice points in Pólya’s walks

In this paper, for any integer $k\ge 2$, we study the distribution of the visible lattice points in certain generalized Pólya walks on $Z^k$: perturbed Pólya walk and twisted Pólya walk. For the first case, we prove that the asymptotic proportion of visible points in a perturbed Pólya walk is almost surely $1/\zeta \left(k\right)$, where $\zeta \left(s\right)$ denotes the Riemann zeta function. A trivial case of our result covers the standard Pólya walk. Moreover, we do numerical experiments for the second case, we conjecture that the proportion is also almost surely $1/\zeta \left(k\right)$.