Scaling limit of random plane quadrangulations with a simple boundary, via restriction

TL;DR

This paper proves that large random quadrangulations with a simple boundary converge to the Brownian disk, using a novel approach of viewing them as conditioned models and applying unconditioning techniques.

Contribution

It introduces a new method to analyze quadrangulations with boundaries by relating them to conditioned models with known scaling limits.

Findings

Quadrangulations with simple boundary converge to the Brownian disk in the Gromov--Hausdorff topology.

The boundary length scales as $p_n o 2eta o 2 ext{constant} imes ext{sqrt}(n)$.

The method of unconditioning can be applied to other models of random maps.

Abstract

We prove that quadrangulations with a simple boundary converge to the Brownian disk. More precisely, we fix a sequence $(p_n)$ of even positive integers with $p_n\sim 2\alpha \sqrt{2n}$ for some $\alpha\in(0,\infty)$. Then, for the Gromov--Hausdorff topology, a quadrangulation with a simple boundary uniformly sampled among those with $n$ inner faces and boundary length $p_n$ weakly converges, in the usual scaling $n^{-1/4}$, toward the Brownian disk of perimeter $3\alpha$. Our method consists in seeing a uniform quadrangulation with a simple boundary as a conditioned version of a model of maps for which the Gromov--Hausdorff scaling limit is known. We then explain how classical techniques of unconditionning can be used in this setting of random maps.