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

2022/05/31 by Mj, Mahan, Roy, Parthanil, Sarkar, Sourav
#60G10 #60G52 #60G60 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA) #Probability (math.PR)

paper · doi:10.48550/arxiv.2205.15849

Abstract

We show that any stationary symmteric 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.

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