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September 2017 The scaling limit of random simple triangulations and random simple quadrangulations
Louigi Addario-Berry, Marie Albenque
Ann. Probab. 45(5): 2767-2825 (September 2017). DOI: 10.1214/16-AOP1124


Let $M_{n}$ be a simple triangulation of the sphere $\mathbb{S}^{2}$, 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 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.


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Louigi Addario-Berry. Marie Albenque. "The scaling limit of random simple triangulations and random simple quadrangulations." Ann. Probab. 45 (5) 2767 - 2825, September 2017.


Received: 1 January 2016; Revised: 1 April 2016; Published: September 2017
First available in Project Euclid: 23 September 2017

zbMATH: 06812193
MathSciNet: MR3706731
Digital Object Identifier: 10.1214/16-AOP1124

Primary: 05C12 , 60F17 , 82B41

Keywords: Brownian map , Brownian snake , Random maps , Spatial branching process

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 5 • September 2017
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