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Local limits of uniform triangulations in high genus

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Abstract

We prove a conjecture of Benjamini and Curien stating that the local limits of uniform random triangulations whose genus is proportional to the number of faces are the planar stochastic hyperbolic triangulations (PSHT) defined in Curien (Probab Theory Relat Fields 165(3):509–540, 2016). The proof relies on a combinatorial argument and the Goulden–Jackson recurrence relation to obtain tightness, and probabilistic arguments showing the uniqueness of the limit. As a consequence, we obtain asymptotics up to subexponential factors on the number of triangulations when both the size and the genus go to infinity. As a part of our proof, we also obtain the following result of independent interest: if a random triangulation of the plane T is weakly Markovian in the sense that the probability to observe a finite triangulation t around the root only depends on the perimeter and volume of t, then T is a mixture of PSHT.

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Notes

  1. As a deterministic example, the d-regular triangulations of the plane for \(d>6\) are hyperbolic.

  2. To be exact, the triangulations defined in  [20] are type-II triangulations, i.e. triangulations with no loop joining a vertex to itself. The type-I (with loops) analogue, which will be the one considered in this work, was defined in  [13].

  3. In particular, if t is a triangulation of a multi-polygon, then \(d_t^*(f,f')\) is the length of the smallest dual path which avoids the external faces.

  4. Unless the number of faces of t is smaller than r, in which case \(B_r^*(f)=t^*\), so any pair of \(F^2\) is good.

  5. This is a “weak” definition of one-endedness, since it does not prevent T to be the dual of a tree. However, once we will have proved that T has finite vertex degrees, this will be equivalent to the usual definition.

  6. In order to have nicer formulas, the convention we use here differs from the rest of the paper, in which the parameter n is related to the total number of vertices. This is why we do not use the \(\tau \) notation. We insist that this holds only in Sect. 3.

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Acknowledgements

The authors thank Guillaume Chapuy and Nicolas Curien for helpful discussions and comments on earlier versions of this manuscript. The authors also thank the two anonymous referees for useful remarks. The first author is supported by ERC Geobrown (740943). The second author is fully supported by ERC-2016-STG 716083 “CombiTop”. The authors would also like to thank the Isaac Newton Institute for Mathematical Sciences (EPSRC Grant No. EP/R014604/1) for its hospitality during the Random Geometry follow-up workshop when this work was started.

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  1. University of British Columbia, Vancouver, Canada

    Thomas Budzinski

  2. IRIF, Université Paris Diderot, Paris, France

    Baptiste Louf

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Budzinski, T., Louf, B. Local limits of uniform triangulations in high genus. Invent. math. 223, 1–47 (2021). https://doi.org/10.1007/s00222-020-00986-3

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