Unimodular hyperbolic triangulations: circle packing and random walk

Omer Angel, Tom Hutchcroft, Asaf Nachmias*, Gourab Ray

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)229-268
Number of pages40
JournalInventiones Mathematicae
Issue number1
StatePublished - 1 Oct 2016


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