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.
"The scaling limit of uniform random plane maps, via the Ambjørn–Budd bijection." Electron. J. Probab. 19 1 - 16, 2014. https://doi.org/10.1214/EJP.v19-3213