Uniqueness and universality of the Brownian map

• Autores: Jean-François Le Gall
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 4, 2013, págs. 2880-2960
• Idioma: inglés
• DOI: 10.1214/12-AOP792
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• Resumen

  • We consider a random planar map Mn which is uniformly distributed over the class of all rooted q-angulations with n faces. We let mn be the vertex set of Mn, which is equipped with the graph distance dgr. Both when q≥4 is an even integer and when q=3, there exists a positive constant cq such that the rescaled metric spaces (mn,cqn−1/4dgr) converge in distribution in the Gromov–Hausdorff sense, toward a universal limit called the Brownian map. The particular case of triangulations solves a question of Schramm.