Journal of the European Mathematical Society
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Poisson boundaries of lamplighter groups: proof of the Kaimanovich–Vershik conjectureRussell Lyons and Yuval Peres (1) Indiana University, Bloomington, USA
(2) Kent State University, USA
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 $\mathbb Z^d (d \geq 3)$ is the Poisson boundary. For $d \geq 5$, this had been shown earlier by Erschler (2011). We extend this to walks of more general types on more general groups.
Keywords: Random walks, free metabelian group, entropy, harmonic functions
Lyons Russell, Peres Yuval: Poisson boundaries of lamplighter groups: proof of the Kaimanovich–Vershik conjecture. J. Eur. Math. Soc. Electronically published on December 7, 2020. doi: 10.4171/JEMS/1030 (to appear in print)