# Convergence of the free Boltzmann quadrangulation with simple boundary to the Brownian disk

## TL;DR

This paper proves that certain random quadrangulations with boundary converge to a continuous limit called the Brownian disk, using a sophisticated topology for curve-decorated metric spaces, and describes the limit of decorated sphere quadrangulations.

## Contribution

It establishes the convergence of free Boltzmann quadrangulations with boundary to the Brownian disk in the GHPU topology, extending understanding of scaling limits of random planar maps.

## Key findings

- Quadrangulations converge to the Brownian disk in GHPU topology.
- Decorated sphere quadrangulations converge to a space formed by gluing two Brownian disks.
- The results connect discrete random maps with continuous random metric spaces.

## Abstract

We prove that the free Boltzmann quadrangulation with simple boundary and fixed perimeter, equipped with its graph metric, natural area measure, and the path which traces its boundary converges in the scaling limit to the free Boltzmann Brownian disk. The topology of convergence is the so-called Gromov-Hausdorff-Prokhorov-uniform (GHPU) topology, the natural analog of the Gromov-Hausdorff topology for curve-decorated metric measure spaces. From this we deduce that a random quadrangulation of the sphere decorated by a $2l$-step self-avoiding loop converges in law in the GHPU topology to the random curve-decorated metric measure space obtained by gluing together two independent Brownian disks along their boundaries.