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The Shark Random Swim

(Lévy Flight with Memory)

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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 \( p=\frac{1}{\alpha } \).

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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.

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  1. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057, Zurich, Switzerland

    Silvia Businger

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  1. Silvia Businger
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Correspondence to Silvia Businger.

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Businger, S. The Shark Random Swim. J Stat Phys 172, 701–717 (2018). https://doi.org/10.1007/s10955-018-2062-5

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  • DOI: https://doi.org/10.1007/s10955-018-2062-5

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