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 (p_n) of even positive integers with p_n ∼ 2α^√2n for some α ∈ (0,∞). 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 3α. 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.