The scaling limit of uniform random plane maps, via the Ambjørn-Budd bijection

We prove that a uniform rooted plane map with n edges converges in distribution after asuitable normalization to the Brownian map for the Gromov-Hausdorff topology. A recent bijection due to Ambjørn and Budd allows to derive this result by a direct coupling with a uniform random quadrangulation with n faces.