Inventiones mathematicae

, Volume 206, Issue 1, pp 229–268 | Cite as

Unimodular hyperbolic triangulations: circle packing and random walk

  • Omer Angel
  • Tom Hutchcroft
  • Asaf Nachmias
  • Gourab Ray


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.



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. All circle packings above were generated using Ken Stephenson’s CirclePack software [40]. We thank Ken for his assistance using this software and for useful conversations. We also thank the referee for their comments and suggestions.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Omer Angel
    • 1
  • Tom Hutchcroft
    • 1
  • Asaf Nachmias
    • 2
  • Gourab Ray
    • 3
  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  3. 3.Statistical LaboratoryUniversity of CambridgeCambridgeUK

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