Skip to main content
SpringerLink
Log in

On the Local Time of the Half-Plane Half-Comb Walk

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

The Half-Plane Half-Comb walk is a random walk on the plane, when we have a square lattice on the upper half-plane and a comb structure on the lower half-plane, i.e., horizontal lines below the x-axis are removed. We prove that the probability that this walk returns to the origin in 2N steps is asymptotically equal to \(2/(\pi N).\) As a consequence, we prove strong laws and a limit distribution for the local time.

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. Andersen, E.S.: On the fluctuations of sums of random variables. Math. Scand. 1, 263–285 (1953)

    Article  MathSciNet  Google Scholar 

  2. Bertacchi, D.: Asymptotic behaviour of the simple random walk on the 2-dimensional comb. Electron J. Probab. 11, 1184–1203 (2006)

    Article  MathSciNet  Google Scholar 

  3. Bertacchi, D., Zucca, F.: Equidistribution of random walks on spheres. J. Stat. Phys. 94, 91–111 (1999)

    Article  MathSciNet  Google Scholar 

  4. Bertacchi, D., Zucca, F.: Uniform asymptotic estimates of transition probabilities on combs. J. Aust. Math. Soc. 75, 325–353 (2003)

    Article  MathSciNet  Google Scholar 

  5. Chen, X.: How often does a Harris recurrent Markov chain recur? Ann. Probab. 27, 1324–1346 (1999)

    Article  MathSciNet  Google Scholar 

  6. Csáki, E., Csörgő, M., Földes, A., Révész, P.: Strong limit theorems for a simple random walk on the 2-dimensional comb. Electr. J. Probab. 14, 2371–2390 (2009)

    MathSciNet  MATH  Google Scholar 

  7. Csáki, E., Csörgő, M., Földes, A., Révész, P.: On the local time of random walk on the 2-dimensional comb. Stoch. Process. Appl. 121, 1290–1314 (2011)

    Article  MathSciNet  Google Scholar 

  8. Csáki, E., Csörgő, M., Földes, A., Révész, P.: Random walk on the half-plane half-comb structure. Ann. Math. Inf. 39, 29–39 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Csáki, E., Csörgő, M., Földes, A., Révész, P.: Strong limit theorems for anisotropic random walks on \({{\mathbb{Z}}^2}\). Period. Math. Hung. 67, 71–94 (2013)

    Article  Google Scholar 

  10. Darling, D.A., Kac, M.: On occupation times for Markoff processes. Trans. Am. Math. Soc. 84, 444–458 (1957)

    Article  MathSciNet  Google Scholar 

  11. Dvoretzky, A., Erdős, P.: Some problems on random walk in space. In: Proceedings on Second Berkeley Symposium, pp. 353–367 (1951)

  12. Erdős, P., Taylor, S.J.: Some problems concerning the structure of random walk paths. Acta Math. Acad. Sci. Hungar. 11, 137–162 (1960)

    Article  MathSciNet  Google Scholar 

  13. Heyde, C.C.: On the asymptotic behaviour of random walks on an anisotropic lattice. J. Stat. Phys. 27, 721–730 (1982)

    Article  Google Scholar 

  14. Heyde, C.C., Westcott, M., Williams, R.J.: The asymptotic behavior of a random walk on a dual-medium lattice. J. Stat. Phys. 28, 375–80 (1982)

    Article  MathSciNet  Google Scholar 

  15. Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Springer, London (1993)

    Book  Google Scholar 

  16. Petrov, V.V.: Limit Theorems of Probability Theory. Sequences of Independent Random Variables, Oxford Studies in Probability, vol. 4. Clarendon Press, Oxford (1995)

    MATH  Google Scholar 

  17. Pitman, J.: Probability. Springer Text in Statistics. Springer, New York (1993)

    Google Scholar 

  18. Révész, P.: Random Walk in Random and Non-random Environments, 3rd edn. World Scientific, Singapore (2013)

    Book  Google Scholar 

  19. Seshadri, V., Lindenberg, K., Schuler, K.E.: Random walks on periodic and random lattices II. Random walk properties via generating function techniques. J. Stat. Phys. 21, 517–548 (1979)

    Article  Google Scholar 

  20. Silver, H., Shuler, K.E., Lindenberg, K.: Two-dimensional anisotropic random walks. In: Statistical Mechanics and Statistical Methods in Theory and Applications Proceeding Symposium, University of Rochester, Rochester, N.Y., 1976, Plenum, New York, pp. 463–505 (1977)

  21. Weiss, G.H., Havlin, S.: Some properties of a random walk on a comb structure. Physica A 134, 474–482 (1986)

    Article  Google Scholar 

Download references

Acknowledgements

We wish to thank the referee of our submission for carefully reading our manuscript and for making a number of helpful suggestions which certainly improved the presentation of this paper.

Author information

Authors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, P.O.B. 127, Budapest, 1364, Hungary

    Endre Csáki

  2. Department of Mathematics, College of Staten Island, CUNY, 2800 Victory Blvd., Staten Island, New York, 10314, USA

    Antónia Földes

Authors
  1. Endre Csáki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antónia Földes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antónia Földes.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Csáki, E., Földes, A. On the Local Time of the Half-Plane Half-Comb Walk. J Theor Probab 35, 1247–1261 (2022). https://doi.org/10.1007/s10959-020-01065-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-020-01065-2

Keywords

Mathematics Subject Classification (2020)

Search

Navigation