Scaling limit of triangulations of polygons

TL;DR

This paper proves that various types of random polygon triangulations, when scaled appropriately, converge to the Brownian disk, advancing understanding of their geometric limits in probabilistic combinatorics.

Contribution

It establishes the convergence of types I, II, and III triangulations with simple boundary to the Brownian disk, using a novel bijection approach.

Findings

Convergence of triangulations to the Brownian disk.

Use of a bijection between triangulations and blossoming forests.

Progress towards the convergence of uniform triangulations under Cardy embedding.

Abstract

We prove that random triangulations of types I, II, and III with a simple boundary under the critical Boltzmann weight converge in the scaling limit to the Brownian disk. The proof uses a bijection due to Poulalhon and Schaeffer between type III triangulations of the $p$-gon and so-called blossoming forests. A variant of this bijection was also used by Addario-Berry and the first author to prove convergence of type III triangulations to the Brownian map, but new ideas are needed to handle the simple boundary. Our result is an ingredient in the program of the second and third authors on the convergence of uniform triangulations under the Cardy embedding.