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Hyperbolic and Parabolic Unimodular Random Maps

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Abstract

We show that for infinite planar unimodular random rooted maps. many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties relating to amenability, conformal geometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation. We also prove that every simply connected unimodular random rooted map is sofic, that is, a Benjamini–Schramm limit of finite maps.

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, Canada

    Omer Angel

  2. Statistical Laboratory, DPMMS, University of Cambridge, Cambridge, UK

    Tom Hutchcroft

  3. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Asaf Nachmias

  4. Department of Mathematics and Statistics, University of Victoria, Victoria, Canada

    Gourab Ray

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  1. Omer Angel
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  2. Tom Hutchcroft
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  3. Asaf Nachmias
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Correspondence to Asaf Nachmias.

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Angel, O., Hutchcroft, T., Nachmias, A. et al. Hyperbolic and Parabolic Unimodular Random Maps. Geom. Funct. Anal. 28, 879–942 (2018). https://doi.org/10.1007/s00039-018-0446-y

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