The scaling limit of random simple triangulations and random simple quadrangulations
| Autor: | Louigi Addario-Berry, Marie Albenque |
|---|---|
| Přispěvatelé: | McGill University = Université McGill [Montréal, Canada], Laboratoire d'informatique de l'École polytechnique [Palaiseau] (LIX), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), ANR-12-JS02-0001,CARTAPLUS,Combinatoire des cartes et applications(2012), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X) |
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Statistics and Probability
82B41 spatial branching process 01 natural sciences Combinatorics 010104 statistics & probability 60F17 05C12 82B41 [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] FOS: Mathematics Mathematics - Combinatorics Brownian snake 0101 mathematics Brownian motion Probability measure Mathematics 010102 general mathematics Winding number Probability (math.PR) Vertex (geometry) [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] Metric space Scaling limit 60F17 Brownian map Random maps Combinatorics (math.CO) Statistics Probability and Uncertainty Closed loop 05C12 Distance Mathematics - Probability |
| Zdroj: | Annals of Probability Annals of Probability, Institute of Mathematical Statistics, 2017, 45 (5), pp.2767-2825. ⟨10.1214/16-AOP1124⟩ Ann. Probab. 45, no. 5 (2017), 2767-2825 |
| ISSN: | 0091-1798 2168-894X |
| DOI: | 10.1214/16-AOP1124⟩ |
| Popis: | Let $M_n$ be a simple triangulation of the sphere $S^2$, drawn uniformly at random from all such triangulations with n vertices. Endow $M_n$ with the uniform probability measure on its vertices. After rescaling graph distance on $V(M_n)$ by $(3/(4n))^{1/4}$, 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 $M_n$. 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. Comment: 47 pages, 10 figures Revised argument in section 6, section 4 rewritten |
| Databáze: | OpenAIRE |
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