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Published July 2018 | public
Journal Article Open

Hyperbolic and Parabolic Unimodular Random Maps


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.

Additional Information

© 2018 Springer. Received: August 12, 2017. Accepted: February 27, 2018. OA was supported by NSERC and the Simons Foundation. TH was supported by a Microsoft Research PhD Fellowship. AN was supported by ISF grant 1207/15, and ERC starting grant 676970 RANDGEOM. GR was supported in part by EPSRC grant EP/I03372X/1. Part of this work was conducted at the Isaac Newton Institute in Cambridge, during the programme 'Random Geometry' supported by EPSRC Grant Number EP/K032208/1.

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Accepted Version - 1612.08693.pdf


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August 19, 2023
August 19, 2023