Alpha-stable random walk has massive thorns

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.