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Journal of Statistical Physics

, Volume 172, Issue 3, pp 701–717 | Cite as

The Shark Random Swim

(Lévy Flight with Memory)
  • Silvia Businger
Article
  • 141 Downloads

Abstract

The Elephant Random Walk (ERW), first introduced by Schütz and Trimper (Phys Rev E 70:045101, 2004), is a one-dimensional simple random walk on \( {\mathbb {Z}} \) having a memory about the whole past. We study the Shark Random Swim, a random walk with memory about the whole past, whose steps are \( \alpha \)-stable distributed with \( \alpha \in (0,2] \). Our aim in this work is to study the impact of the heavy tailed step distributions on the asymptotic behavior of the random walk. We shall see that, as for the ERW, the asymptotic behavior of the Shark Random Swim depends on its memory parameter p, and that a phase transition can be observed at the critical value \( p=\frac{1}{\alpha } \).

Keywords

Random walk with memory Random recursive trees Yule processes 

Notes

Acknowledgements

I would like to thank Jean Bertoin for introducing me to this topic and for his advice and support. I would also like to thank two anonymous referees for their careful reading of an earlier version of this work and their helpful comments.

References

  1. 1.
    Baur, E., Bertoin, J.: The fragmentation process of an infinite recursive tree and Ornstein-Uhlenbeck type processes. Electron. J. Probab 20(98), 1–20 (2015)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Baur, E., Bertoin, J.: Elephant random walks and their connection to Pólya-type urns. Phys. Rev. E 94, 052134 (2016).  https://doi.org/10.1103/PhysRevE.94.052134 ADSCrossRefGoogle Scholar
  3. 3.
    Bercu, B.: A martingale approach for the elephant random walk. J. Phys. A 51(1), 015201 (2018)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bercu, B., Laulin, L.: On the multi-dimensional elephant random walk. arXiv:1709.07345 (2017)
  5. 5.
    Boyer, D., Romo-Cruz, J.C.R.: Solvable random-walk model with memory and its relations with Markovian models of anomalous diffusion. Phys. Rev. E 90, 042136 (2014).  https://doi.org/10.1103/PhysRevE.90.042136 ADSCrossRefGoogle Scholar
  6. 6.
    Chlebus, E.: An approximate formula for a partial sum of the divergent p-series. Appl. Math. Lett. 22, 732–737 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Coletti, C.F., Gava, R., Schütz, G.M.: Central limit theorem for the elephant random walk. J. Math. Phys. 58, 053303 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    da Silva, M.A.A., Cressoni, J.C., Schütz, G.M., Viswanathan, G.M., Trimper, S.: Non-Gaussian propagator for elephant random walks. Phys. Rev. E 88, 022115 (2013).  https://doi.org/10.1103/PhysRevE.88.022115 ADSCrossRefGoogle Scholar
  9. 9.
    Janson, S.: Limit theorems for triangular urn schemes. Probab. Theory Relat. Fields 134(3), 417–452 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Klammler, F., Kimmich, R.: Geometrical restrictions of incoherent transport of water by diffusion in protein or silica fineparticle systems and by flow in a sponge. A study of anomalous properties using an NMR field-gradient technique. Croat. Chem. Acta 65, 455–470 (1992)Google Scholar
  11. 11.
    Kürsten, R.: Random recursive trees and the elephant random walk. Phys. Rev. E 93, 032111 (2016).  https://doi.org/10.1103/PhysRevE.93.032111 ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Mahmoud, H.: Polya Urn Models. Chapman & Hall/CRC, London (2008). ISBN 1420059831, 9781420059830Google Scholar
  13. 13.
    Möhle, M.: The Mittag-Leffler process and a scaling limit for the block counting process of the Bolthausen-Sznitman coalescent. Lat. Am. J. Probab. Math. Stat. 12(1), 35–53 (2015)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Murase, K., Fujiwara, T., Umemura, Y., Suzuki, K., Iino, R., Yamashita, H., Saito, M., Murakoshi, H., Ritchie, K., Kusumi, A.: Ultrafine membrane compartments for molecular diffusion as revealed by single molecule techniques. Biophys. J. 86, 4075–4093 (2004)CrossRefGoogle Scholar
  15. 15.
    Schütz, G.M., Trimper, S.: Elephants can always remember: exact long-range memory effects in a non-Markovian random walk. Phys. Rev. E 70, 045101 (2004).  https://doi.org/10.1103/PhysRevE.70.045101 ADSCrossRefGoogle Scholar
  16. 16.
    Serva, M.: Scaling behavior for random walks with memory of the largest distance from the origin. Phys. Rev. E 88, 052141 (2013).  https://doi.org/10.1103/PhysRevE.88.052141 ADSCrossRefGoogle Scholar
  17. 17.
    Sims, D.W., Southall, E.J., Humphries, N.E., Hays, G.C., Bradshaw, C.J.A., Pitchford, J.W., James, A., Ahmed, M.Z., Brierley, A.S., Hindell, M.A., Morritt, D., Musyl, M.K., Righton, D., Shepard, E.L.C., Wearmouth, V.J., Wilson, R.P., Witt, M.J., Metcalfe, J.D.: Scaling laws of marine predator search behaviour. Nature 451(7182), 1098–1102 (2008)ADSCrossRefGoogle Scholar
  18. 18.
    Wang, K.G.: Long-time-correlation effects and biased anomalous diffusion. Phys. Rev. A 45, 833–837 (1992).  https://doi.org/10.1103/PhysRevA.45.833 ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZurichSwitzerland

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