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Two-Dimensional Random Interlacements and Late Points for Random Walks

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Abstract

We define the model of two-dimensional random interlacements using simple random walk trajectories conditioned on never hitting the origin, and then obtain some properties of this model. Also, for a random walk on a large torus conditioned on not hitting the origin up to some time proportional to the mean cover time, we show that the law of the vacant set around the origin is close to that of random interlacements at the corresponding level. Thus, this new model provides a way to understand the structure of the set of late points of the covering process from a microscopic point of view.

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Authors and Affiliations

  1. Université Paris Diderot-Paris 7, Mathématiques, case 7012, 75205, Paris Cedex 13, France

    Francis Comets

  2. Department of Statistics, Institute of Mathematics, Statistics and Scientific Computation, University of Campinas-UNICAMP, rua Sérgio Buarque de Holanda 651, 13083–859, Campinas, SP, Brazil

    Serguei Popov & Marina Vachkovskaia

Authors
  1. Francis Comets
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  2. Serguei Popov
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  3. Marina Vachkovskaia
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Correspondence to Serguei Popov.

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Communicated by F. Toninelli

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Comets, F., Popov, S. & Vachkovskaia, M. Two-Dimensional Random Interlacements and Late Points for Random Walks. Commun. Math. Phys. 343, 129–164 (2016). https://doi.org/10.1007/s00220-015-2531-5

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  • DOI: https://doi.org/10.1007/s00220-015-2531-5

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