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Span Observables: “When is a Foraging Rabbit No Longer Hungry?”

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Abstract

Be \(X_t\) a random walk. We study its span S, i.e. the size of the domain visited up to time t. We want to know the probability that S reaches 1 for the first time, as well as the density of the span given t. Analytical results are presented, and checked against numerical simulations. We then generalize this to include drift, and one or two reflecting boundaries. We also derive the joint probability of the maximum and minimum of a process. Our results are based on the diffusion propagator with reflecting or absorbing boundaries, for which a set of useful formulas is derived.

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Acknowledgements

After completion of this work we learned that for the drift-free case the time that the span first reaches one was already calculated in Phys. Rev. E 94, 062131 (2016).

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  1. Laboratoire de Physique de l’Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, Paris, France

    Kay Jörg Wiese

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  1. Kay Jörg Wiese
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Correspondence to Kay Jörg Wiese.

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Communicated by Giulio Biroli.

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Wiese, K.J. Span Observables: “When is a Foraging Rabbit No Longer Hungry?”. J Stat Phys 178, 625–643 (2020). https://doi.org/10.1007/s10955-019-02446-6

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  • DOI: https://doi.org/10.1007/s10955-019-02446-6

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