# Local limits of uniform triangulations in high genus

## TL;DR

This paper proves that uniform random triangulations with genus proportional to faces converge locally to PSHT, providing asymptotic counts and characterizing weakly Markovian triangulations as mixtures of PSHT.

## Contribution

It confirms a conjecture linking high-genus triangulations to PSHT and introduces a new characterization of weakly Markovian triangulations.

## Key findings

- Local limits of high-genus triangulations are PSHT.
- Asymptotics for triangulations with growing size and genus.
- Weakly Markovian triangulations are mixtures of PSHT.

## 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 arXiv:1401.3297. 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.