TY - JOUR

T1 - Unimodular hyperbolic triangulations

T2 - circle packing and random walk

AU - Angel, Omer

AU - Hutchcroft, Tom

AU - Nachmias, Asaf

AU - Ray, Gourab

N1 - Publisher Copyright:
© 2016, Springer-Verlag Berlin Heidelberg.

PY - 2016/10/1

Y1 - 2016/10/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84960444433&partnerID=8YFLogxK

U2 - 10.1007/s00222-016-0653-9

DO - 10.1007/s00222-016-0653-9

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AN - SCOPUS:84960444433

SN - 0020-9910

VL - 206

SP - 229

EP - 268

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 1

ER -