Skip to main content
SpringerLink
Log in

Positive Harmonic Functions for Semi-Isotropic Random Walks on Trees, Lamplighter Groups, and DL-Graphs

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

We determine all positive harmonic functions for a large class of “semi-isotropic” random walks on the lamplighter group, i.e., the wreath product ℤq≀ℤ, where q≥2. This is possible via the geometric realization of a Cayley graph of that group as the Diestel–Leader graph \(\mathsf{DL}(q,q)\) . More generally, \(\mathsf{DL}(q,r)\) (q,r≥2) is the horocyclic product of two homogeneous trees with respective degrees q+1 and r+1, and our result applies to all \(\mathsf {DL}\) -graphs. This is based on a careful study of the minimal harmonic functions for semi-isotropic walks on trees.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Babillot, M., Bougerol, Ph. and Elie, L.: ‘The random difference equation X n=A n X n−1+B n in the critical case’, Ann. Probab. 25 (1997), 478–493.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Bertacchi, D.: ‘Random walks on Diestel–Leader graphs’, Abh. Math. Sem. Univ. Hamburg 71 (2001), 205–224.

    Article  MATH  MathSciNet  Google Scholar 

  4. Brofferio, S.: ‘Renewal theory on the affine group of an oriented tree’, J. Theoret. Probab. 17 (2004), 819–859.

    Article  MATH  MathSciNet  Google Scholar 

  5. Brofferio, S. and Woess, W.: ‘Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel–Leader graphs’, Ann. Inst. Poincaré (B) Probab. Statist., to appear.

  6. Cartier, P.: ‘Fonctions harmoniques sur un arbre’, Symposia Math. 9 (1972), 203–270.

    MATH  Google Scholar 

  7. Cartwright, D.I., Kaimanovich, V.A. and Woess, W.: ‘Random walks on the affine group of local fields and of homogeneous trees’, Ann. Inst. Fourier (Grenoble) 44 (1994), 1243–1288.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Diestel, R. and Leader, I.: ‘A conjecture concerning a limit of non-Cayley graphs’, J. Algebraic Combin. 14 (2001), 17–25.

    Article  MathSciNet  MATH  Google Scholar 

  10. Doob, J.L.: ‘Discrete potential theory and boundaries’, J. Math. Mech. 8 (1959), 433–458.

    MATH  MathSciNet  Google Scholar 

  11. Dynkin, E.B.: ‘Boundary theory of Markov processes (the discrete case)’, Russian Math. Surveys 24 (1969), 1–42.

    Article  MATH  Google Scholar 

  12. Elie, L.: ‘Fonctions harmoniques positives sur le groupe affine’, in H. Heyer (ed.), Probability Measures on Groups, Lecture Notes in Math. 706, Springer, Berlin, 1978, pp. 96–110.

    Google Scholar 

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

    MATH  Google Scholar 

  14. Erschler, A.G.: ‘Isoperimetry for wreath products of Markov chains and multiplicity of self-intersections of random walks’, Preprint, Univ. Lille, 2004.

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Hunt, G.A.: ‘Markoff chains and Martin boundaries’, Illinois J. Math. 4 (1960), 313–340.

    MATH  MathSciNet  Google Scholar 

  17. Kaimanovich, V.A.: ‘Poisson boundaries of random walks on discrete solvable groups’, in H. Heyer (ed.), Probability Measures on Groups X, Plenum, New York, 1991, pp. 205–238.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Picardello, M.A. and Woess, W.: ‘Martin boundaries of random walks: Ends of trees and groups’, Trans. Amer. Math. Soc. 302 (1987), 185–205.

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  25. Varopoulos, N.Th.: ‘Théorie du potentiel sur des groupes et des variétés’, C. R. Acad. Sci. Paris Sér. I 302 (1986), 203–205.

    MATH  MathSciNet  Google Scholar 

  26. Woess, W.: ‘Boundaries of random walks on graphs and groups with infinitely many ends’, Israel J. Math. 68 (1989), 271–301.

    MATH  MathSciNet  Google Scholar 

  27. Woess, W.: Random Walks on Infinite Graphs and Groups, Cambridge Tracts in Math. 138, Cambridge University Press, Cambridge, 2000.

    MATH  Google Scholar 

  28. Woess, W.: ‘Lamplighters, Diestel–Leader graphs, random walks, and harmonic functions’, Combin. Probab. Comput. 14 (2005), 415–433.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques, Université Paris-Sud, Bâtiment 425, F-91405, Orsay Cedex, France

    Sara Brofferio

  2. Institut für Mathematik C, Technische Universität Graz, Steyrergasse 30, A-8010, Graz, Austria

    Wolfgang Woess

Authors
  1. Sara Brofferio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wolfgang Woess
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sara Brofferio.

Additional information

Mathematics Subject Classifications (2000)

60J50, 05C25, 20E22, 31C05, 60G50.

Supported by European Commission, Marie Curie Fellowship HPMF-CT-2002-02137.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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). https://doi.org/10.1007/s11118-005-0914-5

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-005-0914-5

Keywords

Search

Navigation