# Scaling limits for random triangulations on the torus

## TL;DR

This paper investigates the scaling limits of random triangulations on the torus, demonstrating convergence of their metric spaces to a limiting random metric space and analyzing associated labelings.

## Contribution

It establishes the Gromov-Hausdorff convergence of scaled random triangulations on the torus and introduces a simple labeling scheme approximating distances.

## Key findings

- Convergence of scaled triangulations to a limiting metric space
- Construction of a simple labeling converging to a scaling limit
- Labeling approximates distances to the root with negligible correction

## Abstract

We study the scaling limit of essentially simple triangulations on the torus. We consider, for every $n\geq 1$, a uniformly random triangulation $G_n$ over the set of (appropriately rooted) essentially simple triangulations on the torus with $n$ vertices. We view $G_n$ as a metric space by endowing its set of vertices with the graph distance denoted by $d_{G_n}$ and show that the random metric space $(V(G_n),n^{-1/4}d_{G_n})$ converges in distribution in the Gromov-Hausdorff sense when $n$ goes to infinity, at least along subsequences, toward a random metric space. One of the crucial steps in the argument is to construct a simple labeling on the map and show its convergence to an explicit scaling limit. We moreover show that this labeling approximates the distance to the root up to a uniform correction of order $o(n^{1/4})$.