The Annals of Probability

The scaling limit of random simple triangulations and random simple quadrangulations

Louigi Addario-Berry and Marie Albenque

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Abstract

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.

Article information

Source
Ann. Probab., Volume 45, Number 5 (2017), 2767-2825.

Dates
Received: January 2016
Revised: April 2016
First available in Project Euclid: 23 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1506132026

Digital Object Identifier
doi:10.1214/16-AOP1124

Mathematical Reviews number (MathSciNet)
MR3706731

Zentralblatt MATH identifier
06812193

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 05C12: Distance in graphs 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Keywords
Random maps Brownian map Brownian snake spatial branching process

Citation

Addario-Berry, Louigi; Albenque, Marie. The scaling limit of random simple triangulations and random simple quadrangulations. Ann. Probab. 45 (2017), no. 5, 2767--2825. doi:10.1214/16-AOP1124. https://projecteuclid.org/euclid.aop/1506132026


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