Alpha-stable random walk has massive thorns

Abstract

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 ${\mathbb{Z}}^d$, $d\ge 3$, 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.

26 pages, 2 figures

Cited by in corpus (3)

• Critical exponents for long-range O(n) models below the upper critical dimension
• Critical two-point function for long-range $O(n)$ models below the upper critical dimension
• Stationary entrance chains and applications to random walks