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...

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Bibliographic Details
Published inGeometric and functional analysis Vol. 28; no. 4; pp. 879 - 942
Main Authors Angel, Omer, Hutchcroft, Tom, Nachmias, Asaf, Ray, Gourab
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.07.2018
Springer Nature B.V
Subjects
Analysis
Curvature
Mathematics
Mathematics and Statistics
Random walk
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Summary: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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 14
ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-018-0446-y