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Tracy-Widom Asymptotics for a River Delta Model

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Stochastic Dynamics Out of Equilibrium (IHPStochDyn 2017)

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Abstract

We study an oriented first passage percolation model for the evolution of a river delta. This model is exactly solvable and occurs as the low temperature limit of the beta random walk in random environment. We analyze the asymptotics of an exact formula from [13] to show that, at any fixed positive time, the width of a river delta of length L approaches a constant times \(L^{2/3}\) with Tracy-Widom GUE fluctuations of order \(L^{4/9}\). This result can be rephrased in terms of particle systems. We introduce an exactly solvable particle system on the integer half line and show that after running the system for only finite time the particle positions have Tracy-Widom fluctuations.

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Acknowledgements

The authors thank Ivan Corwin for many helpful discussions and for useful comments on an earlier draft of the paper. The authors thank an anonymous reviewer for detailed and helpful comments on the manuscript. G. B. was partially supported by the NSF grant DMS:1664650. M. R. was partially supported by the Fernholz Foundation’s “Summer Minerva Fellow” program, and also received summer support from Ivan Corwin’s NSF grant DMS:1811143.

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Authors and Affiliations

  1. Department of Mathematics, Columbia University, 2990 Broadway, New York, NY, 10027, USA

    Guillaume Barraquand & Mark Rychnovsky

Authors
  1. Guillaume Barraquand
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  2. Mark Rychnovsky
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Correspondence to Guillaume Barraquand .

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Editors and Affiliations

  1. Laboratoire de Probabilités, Statistiques et Modélisation (UMR 8001), Université Paris Diderot, Paris, France

    Giambattista Giacomin

  2. CEREMADE, CNRS UMR 7534, Université Paris Dauphine - PSL, Paris, France

    Stefano Olla

  3. CNRS UMR 8145, MAP5, Université Paris Descartes, Paris, France

    Ellen Saada

  4. Zentrum Mathematik, Technische Universität München, Garching, Bayern, Germany

    Herbert Spohn

  5. CERMICS, Ecole des Ponts, Marne-La-Vallée, France

    Gabriel Stoltz

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Barraquand, G., Rychnovsky, M. (2019). Tracy-Widom Asymptotics for a River Delta Model. In: Giacomin, G., Olla, S., Saada, E., Spohn, H., Stoltz, G. (eds) Stochastic Dynamics Out of Equilibrium. IHPStochDyn 2017. Springer Proceedings in Mathematics & Statistics, vol 282. Springer, Cham. https://doi.org/10.1007/978-3-030-15096-9_17

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