Convergence of non-bipartite maps via symmetrization of labeled trees
Annales Henri Lebesgue, Volume 4 (2021) , pp. 653-683.

Metadata

KeywordsRandom trees, Invariance principle, Brownian snake, Random planar maps, Brownian map

Abstract

Fix an odd integer p5. Let M n be a uniform p-angulation with n vertices, endowed with the uniform probability measure on its vertices. We prove that there exists C p + 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.


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