Random walks on products of trees and certain t.d.l.c. groups

TL;DR

This paper investigates the Poisson boundary of products of affine automorphism groups of trees and certain t.d.l.c. groups, establishing conditions for boundary triviality and describing the boundary as a space of ends with a hitting measure.

Contribution

It extends previous work to characterize the Poisson boundary for products of tree automorphism groups and semi-direct products in t.d.l.c. groups, providing new conditions for boundary triviality.

Findings

Poisson boundary is a product of ends of trees with a hitting measure.

Boundary triviality conditions are established for both cases.

Method extends prior work on affine automorphism groups of homogeneous trees.

Abstract

The Poisson boundary of a finite direct product of affine automorphism groups of homogeneous trees is considered. The Poisson boundary is shown to be a product of ends of trees with a hitting measure for spread-out, aperiodic measures of finite first moment whose closed support generates a subgroup which is not fully exceptional. The Poisson boundary of a semi-direct product associated with every automorphism $\alpha$ and tidy compact open subgroup $V$ in a locally compact, totally disconnected group $G$ is also shown to be the space of ends of the tree with the hitting measure under similar assumptions. Necessary and sufficient conditions for boundary triviality are given in both cases. The method of proof is largely an extension of the prior work of Cartwright, Kaimanovich and Woess on affine automorphism groups of homogeneous trees.