The scaling limit of uniform random plane maps, via the Ambj{\o}rn-Budd bijection

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