Alpha-stable random walk has massive thorns

TL;DR

This paper introduces a class of discrete alpha-stable random walks on integer lattices, analyzing conditions under which certain subsets like axes and cones are massive or non-massive, extending continuous stable process theory.

Contribution

It establishes a discrete analogue of symmetric alpha-stable processes and characterizes the massiveness of thorn sets in higher dimensions.

Findings

Axes are non-massive sets for 0<α<2 in dimensions d≥3.

Any cone is a massive set under the same conditions.

Provides necessary and sufficient conditions for thorn sets to be massive.

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