JournalsjemsVol. 23, No. 4pp. 1133–1160

Poisson boundaries of lamplighter groups: proof of the Kaimanovich–Vershik conjecture

  • Russell Lyons

    Indiana University, Bloomington, USA
  • Yuval Peres

    Kent State University, USA
Poisson boundaries of lamplighter groups: proof of the Kaimanovich–Vershik conjecture cover
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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 Zd(d3)\mathbb Z^d (d \geq 3) is the Poisson boundary. For d5d \geq 5, this had been shown earlier by Erschler (2011). We extend this to walks of more general types on more general groups.

Cite this article

Russell Lyons, Yuval Peres, Poisson boundaries of lamplighter groups: proof of the Kaimanovich–Vershik conjecture. J. Eur. Math. Soc. 23 (2021), no. 4, pp. 1133–1160

DOI 10.4171/JEMS/1030