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Two-Dimensional Anisotropic Random Walks: Fixed Versus Random Column Configurations for Transport Phenomena

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Abstract

We consider random walks on the square lattice of the plane along the lines of Heyde (J Stat Phys 27:721–730, 1982, Stochastic processes, Springer, New York, 1993) and den Hollander (J Stat Phys 75:891–918, 1994), whose studies have in part been inspired by the so-called transport phenomena of statistical physics. Two-dimensional anisotropic random walks with anisotropic density conditions á  la Heyde (J Stat Phys 27:721–730, 1982, Stochastic processes, Springer, New York, 1993) yield fixed column configurations and nearest-neighbour random walks in a random environment on the square lattice of the plane as in den Hollander (J Stat Phys 75:891–918, 1994) result in random column configurations. In both cases we conclude simultaneous weak Donsker and strong Strassen type invariance principles in terms of appropriately constructed anisotropic Brownian motions on the plane, with self-contained proofs in both cases. The style of presentation throughout will be that of a semi-expository survey of related results in a historical context.

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Acknowledgements

The Authors wish to thank a number of referees for carefully reading our manuscript, and for their constructive remarks and suggestions that have very much helped us in preparing this revised version of our paper. The research of E. Csáki and P. Révész was supported by Hungarian National Research, Development and Innovation Office - NKFIH K 108615. The research of M. Csörgő was supported by an NSERC Canada Discovery Grant at Carleton University. The research of A. Földes was supported by PSC CUNY Grant No. 69040-0047.

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Authors and Affiliations

  1. A. Rényi Institute of Mathematics, Hung. Acad. Sci., P.O.B. 127, Budapest, 1364, Hungary

    Endre Csáki

  2. School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada

    Miklós Csörgő

  3. Department of Mathematics, College of Staten Islands, CUNY, 2800 Victory Blvd., Staten Island, NY, 10314, USA

    Antónia Földes

  4. Institut für Statistik und Wahrscheinlichkeitstheorie, Technische Universität Wien, Wiedner Hauptstrasse 8-10/107, 1040, Vienna, Austria

    Pál Révész

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  1. Endre Csáki
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  2. Miklós Csörgő
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  3. Antónia Földes
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  4. Pál Révész
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Correspondence to Endre Csáki.

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Csáki, E., Csörgő, M., Földes, A. et al. Two-Dimensional Anisotropic Random Walks: Fixed Versus Random Column Configurations for Transport Phenomena. J Stat Phys 171, 822–841 (2018). https://doi.org/10.1007/s10955-018-2038-5

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  • DOI: https://doi.org/10.1007/s10955-018-2038-5

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