Convergence of odd-angulations via symmetrization of labeled trees

Abstract

Fix $p\ge 5$ 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. In particular, the latter allows to obtain an invariance principle for labeled multitype Galton-Watson trees, with only a weak assumption on the centering of label displacements

25 pages

References in corpus (2)

• The topological structure of scaling limits of large planar maps
• The lineage process in Galton--Watson trees and globally centered discrete snakes

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