Convergence of odd-angulations via symmetrization of labeled trees

Understand

Fix $p\geq 5$ an odd integer integer.

• Let $M_n$ be a uniform $p$-angulation with $n$ vertices and endowed with the uniform probability measure on its vertices.
• We prove that, there exists $C_p\in \mathbb{R}_+$ such that, after rescaling distances by $C_p/n^{1/4}$, $M_n$ converges in distribution for the Gromov-Hausdorff-Prokhorov topology towards the Brownian map.
• To prove the preceding fact, we introduce a `bootstrapping' principle for distributional convergence of random labelled plane trees.