Skip to main content
Log in

A note on the Poisson boundary of lamplighter random walks

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

The main goal of this paper is to determine the Poisson boundary of lamplighter random walks over a general class of discrete groups Γ endowed with a “rich” boundary. The starting point is the Strip Criterion of identification of the Poisson boundary for random walks on discrete groups due to Kaimanovich (Ann. Math. 152:659–692, 2000). A geometrical method for constructing the strip as a subset of the lamplighter group \({\mathbb {Z}_{2}\wr \Gamma}\) starting with a “smaller” strip in the group Γ is developed. Then, this method is applied to several classes of base groups Γ: groups with infinitely many ends, hyperbolic groups in the sense of Gromov, and Euclidean lattices. We show that under suitable hypothesis the Poisson boundary for a class of random walks on lamplighter groups is the space of infinite limit configurations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bartholdi L., Woess W.: Spectral computations on lamplighter groups and Diestel-Leader graphs. J. Fourier Anal. Appl. 11, 175–202 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Brofferio S., Woess W.: Positive harmonic functions for semi-isotropic random walks on trees, lamplighter groups, and DL-graphs. Potential Anal. 24, 245–265 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cartwright D.I., Soardi P.M.: Convergence to ends for random walks on the automorphism group of a tree. Proc. Amer. Math. Soc. 107, 817–823 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  4. Coornaert, M., Delzant, T., Papadopoulos, A.: Géometrié et Theorié des Groupes: les Groupes Hyperboliques de Gromov. Lecture notes in Mathematics, vol. 1441. Springer, Berlin (1990)

  5. Dicks W., Schick Th.: The spectral measure of certain elements of the complex group ring of a wreath product. Geom. Dedicata 93, 121–137 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dicks W., Dunwoody M.J.: Groups Acting on Graphs. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  7. Dunwoody M.J.: Cutting up graphs. Combinatorica 2, 15–23 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  8. Erschler, A.G.: On the asymptotics of the rate of departure to infinity (Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 283, 251–257, 263 (2001)

  9. Erschler A.G.: On drift and entropy growth for random walks on groups. Ann. Probab. 31, 1193–1204 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Freudenthal H.: Über die Enden diskreter Räume und Gruppen. Comment. Math. Helv. 17, 1–38 (1944)

    Article  MathSciNet  Google Scholar 

  11. Furstenberg H.: Random walks and discrete subgroups of Lie groups. In: Ney, P.(eds) Advances in Probability and Related Topics, 1, pp. 1–63. M. Dekker, New York (1971)

    Google Scholar 

  12. Ghys, E., De La Harpe, P. (eds.): Sur les GroupesHyperboliques d’aprés Mikhael Gromov. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  13. Grigorchuk R.I., Zuk A.: The lamplighter group as a group generated by a 2-state automaton, and its spectrum. Geom. Dedicata 87, 209–244 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gromov M.: Hyperbolic groups. In: Gersten, S.M.(eds) Essay in Group Theory, pp. 75–263. Springer, New York (1987)

    Google Scholar 

  15. Kaimanovich V.A.: The Poisson formula for groups with hyperbolic properties. Ann. Math. 152, 659–692 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kaimanovich V.A., Vershik A.M.: Random walks on discrete groups: Boundary and entropy. Ann. Probab. 11, 457–490 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kaimanovich, V.A.: Poisson boundary of discrete groups, unpublished manuscript (2001)

  18. Kaimanovich V.A., Woess W.: Boundary and entropy of space homogeneous Markov Chains. Ann. Probab. 30, 323–363 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Karlsson A., Woess W.: The Poisson boundary of lamplighter random walks on trees. Geom. Dedicata 124, 95–107 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kingman J.: The ergodic theory of subadditive processes. J. Royal Stat. Soc. Ser. B 30, 499–510 (1968)

    MATH  MathSciNet  Google Scholar 

  21. Lyons R., Pemantle R., Peres Y.: Random walks on the lamplighter group. Ann. Probab. 24, 1993–2006 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  22. Pittet, C., Saloff-Coste, L.: Amenable groups, isoperimetric profiles and random walks. In: Geometric Group Theory Down Under (Canberra, 1996), pp. 293–316. de Gruyter, Berlin (1999)

  23. Pittet C., Saloff-Coste L.: On random walks on wreath products. Ann. Probab. 30, 948–977 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  24. Revelle D.: Rate of escape of random walks on wreath products. Ann. Probab. 31, 1917–1934 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. Revelle D.: Heat kernel asymptotics on the lamplighter group. Electron. Comm. Probab. 8, 142–154 (2003)

    MATH  MathSciNet  Google Scholar 

  26. Thomassen C., Woess W.: Vertex-transitive graphs and accessibility. J. Combin. Theory Ser. B 58, 248–268 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  27. Woess W.: Amenable group actions on infinite graphs. Math. Ann. 284, 251–265 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  28. Woess W.: Fixed sets and free subgroups of groups acting on metric spaces. Math. Z. 214, 425–440 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  29. Woess, W.: Random walks on infinite graphs and groups. In: Cambridge Tracts in Mathematics, vol. 138. Cambridge University Press (2000)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ecaterina Sava.

Additional information

Communicated by K. Schmidt.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sava, E. A note on the Poisson boundary of lamplighter random walks. Monatsh Math 159, 379–396 (2010). https://doi.org/10.1007/s00605-009-0103-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-009-0103-5

Keywords

Mathematics Subject Classification (2000)

Navigation