Hyperbolic and Parabolic Unimodular Random Maps
Probability
2018-03-22 v2
Abstract
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.
Cite
@article{arxiv.1612.08693,
title = {Hyperbolic and Parabolic Unimodular Random Maps},
author = {Omer Angel and Tom Hutchcroft and Asaf Nachmias and Gourab Ray},
journal= {arXiv preprint arXiv:1612.08693},
year = {2018}
}
Comments
54 pages, 8 figures. V2: Final version, some minor revisions. To appear in GAFA