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Probability Theory and Related Fields

, Volume 165, Issue 3–4, pp 509–540 | Cite as

Planar stochastic hyperbolic triangulations

  • Nicolas CurienEmail author
Article

Abstract

Pursuing the approach of Angel and Ray (Ann Probab, 2015) we introduce and study a family of random infinite triangulations of the full-plane that satisfy a natural spatial Markov property. These new random lattices naturally generalize Angel and Schramm’s uniform infinite planar triangulation (UIPT) and are hyperbolic in flavor. We prove that they exhibit a sharp exponential volume growth, are non-Liouville, and that the simple random walk on them has positive speed almost surely. We conjecture that these infinite triangulations are the local limits of uniform triangulations whose genus is proportional to the size.

Graphical abstract

An artistic representation of a random (3-connected) triangulation of the plane with hyperbolic flavor.

Mathematics Subject Classification

05C80 05C81 60J10 

Notes

Acknowledgments

I am grateful to Omer Angel, Itai Benjamini, Guillaume Chapuy and Gourab Ray for useful discussions on and around Conjecture 1. Thanks also go to two anonymous referees for a careful reading of this paper.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.CNRS and Université Paris 6ParisFrance

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