TY - JOUR
T1 - Hyperbolic and Parabolic Unimodular Random Maps
AU - Angel, Omer
AU - Hutchcroft, Tom
AU - Nachmias, Asaf
AU - Ray, Gourab
N1 - Publisher Copyright:
© 2018, Springer International Publishing AG, part of Springer Nature.
PY - 2018/7/1
Y1 - 2018/7/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85048789753&partnerID=8YFLogxK
U2 - 10.1007/s00039-018-0446-y
DO - 10.1007/s00039-018-0446-y
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AN - SCOPUS:85048789753
SN - 1016-443X
VL - 28
SP - 879
EP - 942
JO - Geometric and Functional Analysis
JF - Geometric and Functional Analysis
IS - 4
ER -