Skip to main content
SpringerLink
Log in

Multidimensional Walks with Random Tendency

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We introduce a multidimensional walk with memory and random tendency. The asymptotic behaviour is characterized, proving a law of large numbers and showing a phase transition from diffusive to superdiffusive regimes. In first case, we obtain a functional limit theorem to Gaussian vectors. In superdiffusive regime, we obtain strong convergence to a non-Gaussian random vector and characterize its moments.

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.

Similar content being viewed by others

References

  1. Arita, C., Ragoucy, E.: Interacting elephant random walks. Phys. Rev. E 98, 052118 (2018)

    Article  ADS  Google Scholar 

  2. Balenzuela, P., Pinasco, J.P., Semeshenko, V.: The undecided have the key: interaction driven opinion dynamics in a three state mode. PLoS ONE 10, 1 (2015)

    Article  Google Scholar 

  3. Baur, E.: On a class of random walks with reinforced memory. J. Stat. Phys. (2020). https://doi.org/10.1007/s10955-020-02602-3

    Article  MathSciNet  Google Scholar 

  4. Baur, E., Bertoin, J.: Elephant random walks and their connection to Pólya-type urns. Phys. Rev. E 94, 052134 (2016)

    Article  ADS  Google Scholar 

  5. Bercu, B.: A martingale approach for the elephant random walk. J. Phys. A 51(1), 015201 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  6. Bercu, B., Laulin, L.: On the multi-dimensional elephant random walk. J. Stat. Phys. 175, 1146–1163 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  7. Bertenghi, M.: Functional limit theorems for the Multi-dimensional Elephant Random Walk. Preprint arXiv:2004.02004, 2020

  8. Bertoin, J.: Noise reinforcement for Levy processes. Ann. Inst. H. Poincaré Probab. Statist. 56(3), 2236–2252 (2020)

    Article  MathSciNet  Google Scholar 

  9. Bertoin, J.: Universality of Noise Reinforced Brownian Motions. Preprint arXiv:2002.09166, 2020

  10. Bertoin, J.: How linear reinforcement affects Donsker’s theorem for empirical processes. Probab. Theory Relat. Fields (2020). https://doi.org/10.1007/s00440-020-01001-9

    Article  Google Scholar 

  11. Businger, S.: The shark random swim. J. Stat. Phys. 172(3), 701–717 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  12. Coletti, C., Gava, R., Schütz, G.: Central limit theorem for the elephant random walk. J. Math. Phys. 56, 05330 (2017)

    MathSciNet  MATH  Google Scholar 

  13. Cressoni, J., Viswanathan, G., Da Silva, M.: Exact solution of an anisotropic 2D random walk model with strong memory correlations. J. Phys. A 46, 505002 (2013)

    Article  MathSciNet  Google Scholar 

  14. Drezner, Z., Farnum, N.: A generalized binomial distribution. Commun. Statist. Theory Methods 22, 3051–3063 (1993)

    Article  MathSciNet  Google Scholar 

  15. Duflo, M.: Random Iterative Models. Applications of Mathematics, vol. 34. Springer, Berlin (1997)

    Book  Google Scholar 

  16. Galam, S.: The drastic outcomes from voting alliances in three-party democratic voting (1990 \(\rightarrow \) 2013). J. Stat. Phys. 151, 46–48 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  17. González-Navarrete, M., Lambert, R.: Non-Markovian random walks with memory lapses. J. Math. Phys. 59, 113301 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  18. González-Navarrete, M., Lambert, R.: The diffusion of opposite opinions in a randomly biased environment. J. Math. Phys. 60, 113301 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  19. González-Navarrete, M., Lambert, R.: Urn models with two types of strategies. Preprint arXiv:1708.06430, 2019

  20. Gut, A., Stadtmuller, U.: Elephant random walks with delays. Preprint arXiv:1906.04930v1, 2019

  21. Hall, P., Heyde, C.C.: Martingale Limit Theory and Its Application. Academic Press, New York (1980)

    MATH  Google Scholar 

  22. Hod, S., Keshet, U.: Phase transition in random walks with long-range correlations. Phys. Rev. E 70, 015104(R) (2004)

    Article  ADS  Google Scholar 

  23. Janson, S.: Functional limit theorems for multitype branching processes and generalized Pólya urns. Stoch. Proc. Appl. 110(2), 177–245 (2004)

    Article  Google Scholar 

  24. Kubota, N., Takei, M.: Gaussian fluctuation for superdiffusive elephant random walks. J. Stat. Phys. 177, 1157–1171 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  25. Kumar, N., Harbola, U., Lindenberg, K.: Memory-induced anomalous dynamics: emergence of diffusion, subdiffusion, and superdiffusion from a single random walk model. Phys. Rev. E 82, 021101 (2010)

    Article  ADS  Google Scholar 

  26. Mahmoud, H.: Pólya urn Models. CRC Press, Boca Raton (2008)

    Book  Google Scholar 

  27. Marquioni, V.M.: Multi-dimensional elephant random walk with coupled memory. Phys. Rev. E 100, 052131 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  28. Martins, A.C.R.: Discrete opinion dynamics with M choices. Eur. Phys. J. B 91, 1 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  29. Schütz, G., Trimper, S.: Elephants can always remember: exact long-range memory effects in a non-Markovian random walk. Phys. Rev. E 70, 045101 (2004)

    Article  ADS  Google Scholar 

  30. Wu, L., Qi, Y., Yang, J.: Asymptotics for dependent Bernoulli random variables. Statist. Probab. Lett. 82(3), 455–463 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author thanks Rodrigo Lambert and Eugene Pechersky for several comments. This work was partially supported by Fondo Especial DIUBB 1901083-RS from Universidad del Bío-Bío.

Author information

Authors and Affiliations

  1. Departamento de Estadística, Universidad del Bío-Bío, Avda. Collao 1202, CP 4051381, Concepción, Chile

    Manuel González-Navarrete

Authors
  1. Manuel González-Navarrete
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manuel González-Navarrete.

Additional information

Communicated by Gregory Schehr.

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

González-Navarrete, M. Multidimensional Walks with Random Tendency. J Stat Phys 181, 1138–1148 (2020). https://doi.org/10.1007/s10955-020-02621-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-020-02621-0

Keywords

Search

Navigation