Summary: ``Let $\Bbb Z^2$ be the two-dimensional integer lattice. For an integer $k\geq2$, we say a non-zero lattice point in $\Bbb Z^2$ is $k$-full if the greatest common divisor of its coordinates is a $k$-full number. In this paper, we first prove that the density of $k$-full lattice points in $\Bbb Z^2$ is $c_k=\prod_p(1-p^{-2}+p^{-2k})$, where the product runs over all primes. Then we show that the density of $k$-full lattice points on a path of an $\alpha$-random walk in $\Bbb Z^2$ is almost surely $c_k$, which is independent on $\alpha$.''