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Area Distribution for Directed Random Walks

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Abstract

We study the probability distribution for the area under a directed random walk in the plane. The walk can serve as a simple model for avalanches based on the idea that the front of an avalanche can be described by a random walk and the size is given by the area enclosed. This model captures some of the qualitative features of earthquakes, avalanches, and other self-organized critical phenomena in one dimension. By finding nonlinear functional relations for the generating functions we calculate directly the exponent in the size distribution law and find it to be 4/3.

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Author notes

  1. Thordur Jonsson

    Present address: University of Iceland, Dunhaga 3, 107, Reykjavik, Iceland;

Authors and Affiliations

  1. Department of Theoretical Physics, University of Oxford, Oxford, OX1 3NP, United Kingdom

    Thordur Jonsson & John F. Wheater

Authors
  1. Thordur Jonsson
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  2. John F. Wheater
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Jonsson, T., Wheater, J.F. Area Distribution for Directed Random Walks. Journal of Statistical Physics 92, 713–725 (1998). https://doi.org/10.1023/A:1023053024552

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