Convergence of non-bipartite maps via symmetrization of labeled trees

Addario-Berry, Louigi; Albenque, Marie

Annales Henri Lebesgue, Tome 4 (2021), pp. 653-683.

Soient $p\ge 5$ un entier impair fixé et $M_n$ une $p$-angulation uniforme à $n$ sommets, munie de la loi uniforme sur l’ensemble de ses sommets. Nous prouvons qu’il existe une constante $C_p\in {ℝ}_{+}$ telle qu’après avoir renormalisé ses distances par $C_p/n^{1/4}$, $M_n$ converge en distribution vers la carte brownienne au sens de la topologie de Gromov–Hausdorff–Prokhorov. Pour établir ce résultat, nous introduisons une technique de bootstrap qui permet d’obtenir la converence d’arbres étiquetés aléatoires. En particulier, celle-ci nous permet d’obtenir un résultat d’invariance pour la convergence d’arbres de Galton–Watson multitypes étiquetés en ne supposant que des hypothèses faibles sur la distribution de leurs étiquettes.

Fix an odd integer $p\ge 5$. 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 $C_p\in {ℝ}_{+}$ 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.

Classification : 05C10, 05C12, 60C05, 60F17, 60J80
Mots clés : Random trees, Invariance principle, Brownian snake, Random planar maps, Brownian map

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