Hyperbolic and Parabolic Unimodular Random Maps

Omer Angel, Tom Hutchcroft, Asaf Nachmias*, Gourab Ray

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

25 Scopus citations


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.

Original languageEnglish
Pages (from-to)879-942
Number of pages64
JournalGeometric and Functional Analysis
Issue number4
StatePublished - 1 Jul 2018


FundersFunder number
Simons Foundation
Microsoft Research
Horizon 2020 Framework Programme676970
Natural Sciences and Engineering Research Council of Canada
Engineering and Physical Sciences Research CouncilEP/I03372X/1, EP/K032208/1
European Research Council
Israel Science Foundation1207/15


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