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

We prove that quadrangulations with a simple boundary converge to the Brownian disk. More precisely, we fix a sequence / (pn) of even positive integers with pn similar to 2 alpha 2n for some alpha is an element of (0, infinity). Then, for the Gromov-Hausdorff topology, a quadrangulation with a simple boundary uniformly sampled among those with n inner faces and boundary length pn 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.