Mixing Properties of Stable Random Fields Indexed by Amenable and Hyperbolic Groups

Abstract

We show that any stationary symmetric α-stable (SαS) random field indexed by a countable amenable group G is weakly mixing if and only if it is generated by a null action, extending works of Samorodnitsky and Wang-Roy-Stoev for abelian groups to all amenable groups. This enables us to improve significantly the domain of a recently discovered connection to von Neumann algebras. We also establish ergodicity of stationary SαS fields associated with boundary and double boundary actions of a hyperbolic group G, where the boundary is equipped with either the Patterson-Sullivan or the hitting measure of a random walk, and the double boundary is equipped with the Bowen-Margulis-Sullivan measure.

Symmetric stable processes on amenable groups

“…After this work was first published, I was informed by M. Mj, P. Roy and S. Sarkar that they had also been working on a proof of Theorem 1.3. Their results can be found in [37].…”