Local limits of uniform triangulations with boundaries in high genus

TL;DR

This paper investigates the local limits of uniform triangulations with boundaries in high genus, revealing new hyperbolic triangulation limits and providing a combinatorial approach that extends previous results.

Contribution

It introduces the first construction of hyperbolic half-plane triangulations as local limits of large genus triangulations and simplifies the proof of known limits using combinatorial estimates.

Findings

Local limits around boundary edges are hyperbolic half-plane triangulations.

When p=o(n), the local limit rooted on a random edge is the PSHT.

The proof avoids complex recurrence relations, enabling broader applicability.

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.