Unimodular hyperbolic triangulations: circle packing and random walk
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Peer-reviewed
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Abstract
We show that the circle packing type of a unimodular random plane triangulation is parabolic if and only if the expected degree of the root is six, if and only if the triangulation is amenable in the sense of Aldous and Lyons [1]. As a part of this, we obtain an alternative proof of the Benjamini–Schramm Recurrence Theorem [19]. Secondly, in the hyperbolic case, we prove that the random walk almost surely converges to a point in the unit circle, that the law of this limiting point has full support and no atoms, and that the unit circle is a realisation of the Poisson boundary. Finally, we show that the simple random walk has positive speed in the hyperbolic metric.
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Keywords
4901 Applied Mathematics, 49 Mathematical Sciences, 4904 Pure Mathematics, 4905 Statistics
Journal Title
Inventiones Mathematicae
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Journal ISSN
0020-9910
1432-1297
1432-1297
Volume Title
206
Publisher
Springer Science and Business Media LLC
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All rights reserved
Sponsorship
Engineering and Physical Sciences Research Council (EP/I03372X/1)
OA is supported in part by NSERC. AN is supported by the Israel Science Foundation Grant 1207/15 as well as NSERC and NSF grants. GR is supported in part by the Engineering and Physical Sciences Research Council under Grant EP/103372X/1.