Hyperbolic and Parabolic Unimodular Random Maps

By Omer Angel, Tom Hutchcroft, Asaf Nachmias, Gourab Ray. Geometric and Functional Analysis, 2016

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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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bd988a5c20bb66dccd8e5c52386f67ede13f4c27