Electronic Journal of Probability

Recurrence and transience of a multi-excited random walk on a regular tree

Anne-Laure Basdevant and Arvind Singh

Full-text: Open access

Abstract

We study a model of multi-excited random walk on a regular tree which generalizes the models of the once excited random walk and the digging random walk introduced by Volkov (2003). We show the existence of a phase transition and provide a criterion for the recurrence/transience property of the walk. In particular, we prove that the asymptotic behaviour of the walk depends on the order of the excitations, which contrasts with the one dimensional setting studied by Zerner (2005). We also consider the limiting speed of the walk in the transient regime and conjecture that it is not a monotonic function of the environment.

Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 55, 1628-1669.

Dates
Accepted: 9 July 2009
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819516

Digital Object Identifier
doi:10.1214/EJP.v14-672

Mathematical Reviews number (MathSciNet)
MR2525106

Zentralblatt MATH identifier
1203.60143

Subjects
Primary: 60F20: Zero-one laws
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Multi-excited random walk self-interacting random walk branching Markov chain

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Basdevant, Anne-Laure; Singh, Arvind. Recurrence and transience of a multi-excited random walk on a regular tree. Electron. J. Probab. 14 (2009), paper no. 55, 1628--1669. doi:10.1214/EJP.v14-672. https://projecteuclid.org/euclid.ejp/1464819516


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References

  • Amir, G.; Benjamini, I. Kozma, G. Excited random walk against a wall. Probab. Theory Related Fields 140 (2008), no. 1-2, 83–102.
  • Antal, T.; Redner, S. The excited random walk in one dimension. J. Phys. A 38 (2005), no. 12, 2555–2577.
  • Athreya, K. B.; Ney, P. E. Branching processes. Die Grundlehren der mathematischen Wissenschaften, Band 196. Springer-Verlag, New York-Heidelberg, 1972. xi+287 pp.
  • Basdevant, A.-L.; Singh, A.. On the speed of a cookie random walk. Probab. Theory Related Fields 141 (2008), no. 3-4, 625–645.
  • Basdevant, A.-L.; Singh, A.. Rate of growth of a transient cookie random walk. Electron. J. Probab. 13 (2008), no. 26, 811–851.
  • Benjamini, I.; Wilson, D. B. Excited random walk. Electron. Comm. Probab. 8 (2003), 86–92 (electronic).
  • Bérard, J.; Ramírez, A.. Central limit theorem for the excited random walk in dimension D≥2. Electron. Comm. Probab. 12 (2007), 303–314 (electronic).
  • Bertoin, J.. Lévy processes. Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. x+265 pp. ISBN: 0-521-56243-0
  • Chi, Z.. Limit laws of estimators for critical multi-type Galton-Watson processes. Ann. Appl. Probab. 14 (2004), no. 4, 1992–2015.
  • Harris, Th. E. The theory of branching processes. Die Grundlehren der Mathematischen Wissenschaften, Bd. 119 Springer-Verlag, Berlin; Prentice-Hall, Inc., Englewood Cliffs, N.J. 1963 xiv+230 pp.
  • Kosygina, E.; Zerner, M. P. W. Positively and negatively excited random walks on integers, with branching processes. Electron. J. Probab. 13 (2008), no. 64, 1952–1979.
  • Kozma, G. Excited random walk in three dimensions has positive speed. Preprint (2003), math.PR/0310305
  • Kozma, G. Excited random walk in two dimensions has linear speed. Preprint (2005), math.PR/0512535
  • Lyons, R.; Pemantle, R.; Peres, Y.. Biased random walks on Galton-Watson trees. Probab. Theory Related Fields 106 (1996), no. 2, 249–264.
  • Lyons, R.; Pemantle, R.; Peres, Y.. Unsolved problems concerning random walks on trees. Classical and modern branching processes (Minneapolis, MN, 1994), 223–237, IMA Vol. Math. Appl., 84, Springer, New York, 1997.
  • Menshikov, M. V.; Volkov, S. E. Branching Markov chains: qualitative characteristics. Markov Process. Related Fields 3 (1997), no. 2, 225–241.
  • Mountford, T.; Pimentel, L. P. R.; Valle, G.. On the speed of the one-dimensional excited random walk in the transient regime. ALEA Lat. Am. J. Probab. Math. Stat. 2 (2006), 279–296 (electronic).
  • Müller, S.. Recurrence for branching Markov chains. Electron. Commun. Probab. 13 (2008), 576–605.
  • Seneta, E. Nonnegative matrices and Markov chains. Second edition. Springer Series in Statistics. Springer-Verlag, New York, 1981. xiii+279 pp. ISBN: 0-387-90598-7
  • Seneta, E.; Vere-Jones, D. On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Probability 3 1966 403–434.
  • van der Hofstad, R.; Holmes, M. An expansion for self-interacting random walks. Preprint (2007), abs/0706.0614v3
  • van der Hofstad, R.; Holmes, M. Monotonicity for excited random walk in high dimensions. To appear in Probab. Theory Related Fields (2009), abs/0803.1881
  • Vere-Jones, D. Ergodic properties of nonnegative matrices. I. Pacific J. Math. 22 1967 361–386.
  • Volkov, S.. Excited random walk on trees. Electron. J. Probab. 8 (2003), no. 23, 15 pp. (electronic).
  • Zerner, M. P. W. Multi-excited random walks on integers. Probab. Theory Related Fields 133 (2005), no. 1, 98–122.
  • Zerner, M. P. W. Recurrence and transience of excited random walks on Zd and strips. Electron. Comm. Probab. 11 (2006), 118–128 (electronic).