The scaling limit of random simple triangulations and random simple quadrangulations

• Louigi Addario-Berry ^[1] ; Marie Albenque ^[2]
  1. [1] McGill University

     McGill University

     Canadá

     
  2. [2] Centre National de la Recherche Scientifique

     Centre National de la Recherche Scientifique

     París, Francia

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 5, 2017, págs. 2767-2825
• Idioma: inglés
• DOI: 10.1214/16-AOP1124
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• Resumen

  • Let MnMn be a simple triangulation of the sphere S2S2, drawn uniformly at random from all such triangulations with nn vertices. Endow MnMn with the uniform probability measure on its vertices. After rescaling graph distance by (3/(4n))1/4(3/(4n))1/4, the resulting random measured metric space converges in distribution, in the Gromov–Hausdorff–Prokhorov sense, to the Brownian map. In proving the preceding fact, we introduce a labelling function for the vertices of MnMn. Under this labelling, distances to a distinguished point are essentially given by vertex labels, with an error given by the winding number of an associated closed loop in the map. We establish similar results for simple quadrangulations.