  We introduce and study a class of random walks defined on the integer lattice
$ \mathbb{Z} ^d$ -- a discrete space and time counterpart of the symmetric
$\alpha$-stable process in $\mathbb{R} ^d$. When $0< \alpha <2$ any coordinate
axis in $\mathbb{Z} ^d$, $d\geq 3$, is a non-massive set whereas any cone is
massive. We provide a necessary and sufficient condition for the thorn to be a
massive set.
