  Let $\mathbb{Z}^2$ be the two-dimensional integer lattice. For an integer
$k\geq 1$, a non-zero lattice point is $k$-free if the greatest common divisor
of its coordinates is a $k$-free number. We consider the proportions of
$k$-free and twin $k$-free lattice points on a path of an $\alpha$-random
walker in $\mathbb{Z}^2$. Using the second-moment method and tools from
analytic number theory, we prove that these two proportions are $1/\zeta(2k)$
and $\prod_{p}(1-2p^{-2k})$, respectively, where $\zeta$ is the Riemann zeta
function and the infinite product takes over all primes.
