Poisson Boundaries of Lamplighter Groups: Proof of the Kaimanovich-Vershik Conjecture

TL;DR

This paper proves that the final configuration of lamps in simple random walks on lamplighter groups over ${f Z}^d$ (for $d eq 4$) is the Poisson boundary, confirming a long-standing conjecture and extending previous results.

Contribution

It confirms the Kaimanovich-Vershik conjecture for a broad class of lamplighter groups and random walks, generalizing earlier partial results.

Findings

Final configuration of lamps equals the Poisson boundary for $d eq 4$

Extension of results to more general groups and walk types

Resolution of the 1979 conjecture by Kaimanovich and Vershik

Abstract

We answer positively a question of Kaimanovich and Vershik from 1979, showing that the final configuration of lamps for simple random walk on the lamplighter group over ${\Bbb Z}^d$ ($d \ge 3$) is the Poisson boundary. For $d \ge 5$, this had been shown earlier by Erschler (2011). We extend this to walks of more general types on more general groups.