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Local limits of uniform triangulations with boundaries in high genus

Tanguy Lions

Abstract

We study the local limits of uniform random triangulations with boundaries in the regime where the genus is proportional to the number of faces. Budzinski and Louf proved in 2020 that when there are no boundaries, the local limits exist and are the Planar Stochastic Hyperbolic Triangulation (PSHT) introduced in PSHT. We show that when the triangulations considered have size n and boundaries with total length p that tends to infinity with n and p=o(n), the local limits around a typical boundary edge are the half-plane hyperbolic triangulations defined by Angel and Ray. This provides, for the first time, a construction of these hyperbolic half-plane triangulations as local limits of large genus triangulations. We also prove that under the condition p = o(n), the local limit when rooted on a uniformly chosen oriented edge is given by the PSHT. Contrary to the proof of Budzinski and Louf, the latter does not rely on the Goulden-Jackson recurrence relation, but only on coarse combinatorial estimates. Thus, we expect that the proof can be adapted to local limits in similar models.

Local limits of uniform triangulations with boundaries in high genus

Abstract

We study the local limits of uniform random triangulations with boundaries in the regime where the genus is proportional to the number of faces. Budzinski and Louf proved in 2020 that when there are no boundaries, the local limits exist and are the Planar Stochastic Hyperbolic Triangulation (PSHT) introduced in PSHT. We show that when the triangulations considered have size n and boundaries with total length p that tends to infinity with n and p=o(n), the local limits around a typical boundary edge are the half-plane hyperbolic triangulations defined by Angel and Ray. This provides, for the first time, a construction of these hyperbolic half-plane triangulations as local limits of large genus triangulations. We also prove that under the condition p = o(n), the local limit when rooted on a uniformly chosen oriented edge is given by the PSHT. Contrary to the proof of Budzinski and Louf, the latter does not rely on the Goulden-Jackson recurrence relation, but only on coarse combinatorial estimates. Thus, we expect that the proof can be adapted to local limits in similar models.
Paper Structure (32 sections, 35 theorems, 277 equations, 30 figures)

This paper contains 32 sections, 35 theorems, 277 equations, 30 figures.

Table of Contents

  1. Introduction
  2. Enumeration of maps.
  3. Random maps.
  4. Random maps in large genus.
  5. Planar/half-planar stochastic hyperbolic triangulations.
  6. Uniform triangulations with boundaries.
  7. Motivation: long-term exploration of uniform triangulations.
  8. Strategy of the proof.
  9. Acknowledgements.
  10. Preliminaries
  11. Definitions
  12. Combinatorics
  13. Local convergence and dual local convergence
  14. Triangulations of the plane
  15. Triangulations of the half-plane
  16. ...and 17 more sections

Key Result

Theorem 1.1

Fix $\theta \in [0,\frac{1}{2})$, $\frac{g_n}{n} \underset{n \to +\infty}{\longrightarrow}\theta$ and $\mathbf{p}^{n} = (p^n_{1},\cdots,p^n_{\ell_n})$ such that $|\mathbf{p}^{n}| = o(n)$ and $\frac{|\mathbf{p}^{n}|}{\ell_n} \underset{n \to +\infty}{\longrightarrow}+\infty$. Denote by $e^n$ a unifor where $\lambda(\theta)$ is the unique solution to the equation

Figures (30)

  • Figure 1: On the left: a realisation of $\mathbb{H}_{\lambda}$. On the right, a triangulation $t$ with one infinite boundary (dark grey face) and one infinite hole (white face) such that $t \subset \mathbb{H}_{\lambda}$. On this example, $|t_{\mathrm{in}}| = 17$ denotes the number of green vertices and $|\partial^{*}t|-|\partial t| = 8 - 5 = 3$ denotes the number of red edges minus the number of blue edges.
  • Figure 2: The red neighbourhood of the edge $e^n$ touches another boundary and the opposite side of the boundary $\partial_i$. These are the two pathological cases to exclude.
  • Figure 3: Left: a triangulation of the $(6,8,9,14)$-gon of genus $5$ with $n$ vertices. Right: a triangulation with holes $t_0 \subset t$, which has $1$ boundary and $5$ holes. Triangulations of multi-polygons $t_1$ and $t_2$ are the connected components of $t \backslash t_0$. The triangulation $t_1$ (resp. $t_2$) is a triangulation of the $(6,8,5,13)$-gon (resp. $(14,2,4,7)$-gon).
  • Figure 4: We represent a triangulation $t$ of the $\mathbf{p}$-gon. On the left, we show the ball of radius $2$ centered at $e_1$, which lies on a boundary, and at $e_2$, which lies far from the boundaries. On the right, we show the dual local ball of radius $2$ centered at $e_1$, $e_2$, and $e_3$. Note that $B_2^{*}(t,e_1)$ contains the boundary on which $e_1$ lies. Moreover, in $B^{*}_1(t,e_3)$, one of the green internal faces shares a vertex with a boundary face, and thus, this boundary face also belongs to $B^{*}_2(t,e)$.
  • Figure 5: On the left, the triangulation $\mathbb{T}_0$. On the right, the triangulation $\mathbb{T}_{\star}$.
  • ...and 25 more figures

Theorems & Definitions (82)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • ...and 72 more