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

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.