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Journal of Theoretical Probability

, Volume 24, Issue 1, pp 118–149 | Cite as

Random Walks on Directed Covers of Graphs

  • Lorenz A. Gilch
  • Sebastian Müller
Article

Abstract

Directed covers of finite graphs are also known as periodic trees or trees with finitely many cone types. We expand the existing theory of directed covers of finite graphs to those of infinite graphs. While the lower growth rate still equals the branching number, upper and lower growth rates no longer coincide in general. Furthermore, the behavior of random walks on directed covers of infinite graphs is more subtle. We provide a classification in terms of recurrence and transience and point out that the critical random walk may be recurrent or transient. Our proof is based on the observation that recurrence of the random walk is equivalent to the almost sure extinction of an appropriate branching process. Two examples in random environment are provided: homesick random walk on infinite percolation clusters and random walk in random environment on directed covers. Furthermore, we calculate, under reasonable assumptions, the rate of escape with respect to suitable length functions and prove the existence of the asymptotic entropy providing an explicit formula which is also a new result for directed covers of finite graphs. In particular, the asymptotic entropy of random walks on directed covers of finite graphs is positive if and only if the random walk is transient.

Keywords

Trees Random walk Recurrence Transience Upper Collatz–Wielandt number Branching process Rate of escape Asymptotic entropy 

Mathematics Subject Classification (2000)

05C05 60J10 60F05 60J85 

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References

  1. 1.
    Avez, A.: Entropie des groupes de type fini. C. R. Acad. Sci. Paris Sér. A–B 275, A1363–A1366 (1972) MathSciNetGoogle Scholar
  2. 2.
    Benjamini, I., Peres, Y.: Tree-indexed random walks on groups and first passage percolation. Probab. Theory Relat. Fields 98(1), 91–112 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bertacchi, D., Zucca, F.: Characterization of critical values of branching random walks on weighted graphs through infinite-type branching processes. J. Stat. Phys. 134(1), 53–65 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Björklund, M.: Central limit theorems for Gromov hyperbolic groups. J. Theor. Probab. doi: 10.1007/s10959-009-0230-x (2009, to appear)
  5. 5.
    Blachère, S., Brofferio, S.: Internal diffusion limited aggregation on discrete groups having exponential growth. Preprint available at arXiv:math/0507582v1 [math.PR] (2005)
  6. 6.
    Blachère, S., Haïssinsky, P., Mathieu, P.: Asymptotic entropy and Green speed for random walks on countable groups. Ann. Probab. 36(3), 1134–1152 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cover, T., Thomas, J.: Elements of Information Theory, 2nd edn. Wiley, New York (2006) zbMATHGoogle Scholar
  8. 8.
    Derriennic, Y.: Quelques applications du théorème ergodique sous-additif. Astérisque 74, 183–201 (1980) zbMATHMathSciNetGoogle Scholar
  9. 9.
    Doyle, P.G., Snell, J.L.: Random Walks and Electric Networks. Carus Mathematical Monographs, vol. 22. Mathematical Association of America, Washington (1984) zbMATHGoogle Scholar
  10. 10.
    Fayolle, G., Malyshev, V.A., Menshikov, M.V.: Topics in the Constructive Theory of Countable Markov Chains. Cambridge University Press, Cambridge (1995) zbMATHGoogle Scholar
  11. 11.
    Förster, K.-H., Nagy, B.: Local spectral radii and Collatz–Wielandt numbers of monic operator polynomials with nonnegative coefficients. Linear Algebra Appl. 268, 41–57 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gairat, A.S., Malyshev, V.A., Menshikov, M.V., Pelikh, K.D.: Classification of Markov chains that describe the evolution of random strings. Usp. Mat. Nauk 50(2(302)), 5–24 (1995) MathSciNetGoogle Scholar
  13. 13.
    Gantert, N., Müller, S., Popov, S., Vachkovskaia, M.: Survival of branching random walks in random environment. J. Theor. Probab. doi: 10.1007/s10959-009-0227-5 (2009, to appear)
  14. 14.
    Gilch, L.: Rate of escape of random walks on free products. J. Aust. Math. Soc. 83(1), 31–54 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kaimanovich, V.: Hausdorff dimension of the harmonic measure on trees. Ergod. Theory Dyn. Syst. 18, 631–660 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kaimanovich, V.A., Vershik, A.M.: Random walks on discrete groups: boundary and entropy. Ann. Probab. 11, 457–490 (1983) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Lyons, R.: Random walks and percolation on trees. Ann. Probab. 18, 931–958 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lyons, R., Peres, Y.: Probability on Trees and Networks. Current version available at http://mypage.iu.edu/~rdlyons/ (2009, in preparation)
  19. 19.
    Müller, S.: A criterion for transience of multidimensional branching random walk in random environment. Electron. J. Probab. 13, 1189–1202 (2008) zbMATHMathSciNetGoogle Scholar
  20. 20.
    Nagnibeda, T., Woess, W.: Random walks on trees with finitely many cone types. J. Theor. Probab. 15(2), 383–422 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Schaefer, H.H.: Banach Lattices and Positive Operators. Die Grundlehren der Mathematischen Wissenschaften, Band 215. Springer, New York (1974) zbMATHGoogle Scholar
  22. 22.
    Seneta, E.: Non-negative Matrices and Markov Chains. Springer Series in Statistics. Springer, New York (2006). Revised reprint of the second (1981) edition [Springer, New York; MR0719544] zbMATHGoogle Scholar
  23. 23.
    Takacs, C.: Random walk on periodic trees. Electron. J. Probab. 2, 1–16 (1997) MathSciNetGoogle Scholar
  24. 24.
    von Below, J.: An index theory for uniformly locally finite graphs. Linear Algebra Appl. 431, 1–19 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics, vol. 138. Cambridge University Press, Cambridge (2000) zbMATHCrossRefGoogle Scholar
  26. 26.
    Zeitouni, O.: Random walks in random environment. In: Lectures on Probability Theory and Statistics. Lecture Notes in Math., vol. 1837, pp. 189–312. Springer, Berlin (2004) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institut für Mathematische StrukturtheorieGraz University of TechnologyGrazAustria

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