Advertisement

Potential Analysis

, Volume 24, Issue 3, pp 245–265 | Cite as

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

  • Sara Brofferio
  • Wolfgang Woess
Article

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.

Keywords

lamplighter group wreath product Diestel–Leader graph random walk minimal harmonic functions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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. CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Bartholdi, L. and Woess, W.: ‘Spectral computations on lamplighter groups and Diestel–Leader graphs’, J. Fourier Anal. Appl. 11 (2005), 175–202. CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Bertacchi, D.: ‘Random walks on Diestel–Leader graphs’, Abh. Math. Sem. Univ. Hamburg 71 (2001), 205–224. MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Brofferio, S.: ‘Renewal theory on the affine group of an oriented tree’, J. Theoret. Probab. 17 (2004), 819–859. CrossRefMATHMathSciNetGoogle Scholar
  5. 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. Google Scholar
  6. 6.
    Cartier, P.: ‘Fonctions harmoniques sur un arbre’, Symposia Math. 9 (1972), 203–270. MATHGoogle Scholar
  7. 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. MathSciNetMATHGoogle Scholar
  8. 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. CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Diestel, R. and Leader, I.: ‘A conjecture concerning a limit of non-Cayley graphs’, J. Algebraic Combin. 14 (2001), 17–25. CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Doob, J.L.: ‘Discrete potential theory and boundaries’, J. Math. Mech. 8 (1959), 433–458. MATHMathSciNetGoogle Scholar
  11. 11.
    Dynkin, E.B.: ‘Boundary theory of Markov processes (the discrete case)’, Russian Math. Surveys 24 (1969), 1–42. CrossRefMATHGoogle Scholar
  12. 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. 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). MATHGoogle Scholar
  14. 14.
    Erschler, A.G.: ‘Isoperimetry for wreath products of Markov chains and multiplicity of self-intersections of random walks’, Preprint, Univ. Lille, 2004. Google Scholar
  15. 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. CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Hunt, G.A.: ‘Markoff chains and Martin boundaries’, Illinois J. Math. 4 (1960), 313–340. MATHMathSciNetGoogle Scholar
  17. 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. 18.
    Kaimanovich, V.A. and Vershik, A.M.: ‘Random walks on discrete groups: Boundary and entropy’, Ann. Probab. 11 (1983), 457–490. MathSciNetMATHGoogle Scholar
  19. 19.
    Lyons, R., Pemantle, R. and Peres, Y.: ‘Random walks on the lamplighter group’, Ann. Probab. 24 (1996), 1993–2006. CrossRefMathSciNetMATHGoogle Scholar
  20. 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. CrossRefMathSciNetMATHGoogle Scholar
  21. 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. 22.
    Pittet, C. and Saloff-Coste, L.: ‘On random walks on wreath products’, Ann. Probab. 30 (2002), 948–977. CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Revelle, D.: ‘Rate of escape of random walks on wreath products’, Ann. Probab. 31 (2003), 1917–1934. CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Revelle, D.: ‘Heat kernel asymptotics on the lamplighter group’, Electron. Comm. Probab. 8 (2003), 142–154. MATHMathSciNetGoogle Scholar
  25. 25.
    Varopoulos, N.Th.: ‘Théorie du potentiel sur des groupes et des variétés’, C. R. Acad. Sci. Paris Sér. I 302 (1986), 203–205. MATHMathSciNetGoogle Scholar
  26. 26.
    Woess, W.: ‘Boundaries of random walks on graphs and groups with infinitely many ends’, Israel J. Math. 68 (1989), 271–301. MATHMathSciNetGoogle Scholar
  27. 27.
    Woess, W.: Random Walks on Infinite Graphs and Groups, Cambridge Tracts in Math. 138, Cambridge University Press, Cambridge, 2000. MATHGoogle Scholar
  28. 28.
    Woess, W.: ‘Lamplighters, Diestel–Leader graphs, random walks, and harmonic functions’, Combin. Probab. Comput. 14 (2005), 415–433. MATHMathSciNetGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Laboratoire de MathématiquesUniversité Paris-SudOrsay CedexFrance
  2. 2.Institut für Mathematik CTechnische Universität GrazGrazAustria

Personalised recommendations