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.

This is the updated version of the paper previously entitled "Nonbijective scaling limit of maps via restriction."