Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Scaling limit of random planar quadrangulations with a boundary

Jérémie Bettinelli

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Abstract

We discuss the scaling limit of large planar quadrangulations with a boundary whose length is of order the square root of the number of faces. We consider a sequence $(\sigma_{n})$ of integers such that $\sigma_{n}/\sqrt{2n}$ tends to some $\sigma\in[0,\infty]$. For every $n\ge1$, we denote by $\mathfrak{q}_{n}$ a random map uniformly distributed over the set of all rooted planar quadrangulations with a boundary having $n$ faces and $2\sigma_{n}$ half-edges on the boundary. For $\sigma\in(0,\infty)$, we view $\mathfrak{q}_{n}$ as a metric space by endowing its set of vertices with the graph metric, rescaled by $n^{-1/4}$. We show that this metric space converges in distribution, at least along some subsequence, toward a limiting random metric space, in the sense of the Gromov–Hausdorff topology. We show that the limiting metric space is almost surely a space of Hausdorff dimension $4$ with a boundary of Hausdorff dimension $2$ that is homeomorphic to the two-dimensional disc. For $\sigma=0$, the same convergence holds without extraction and the limit is the so-called Brownian map. For $\sigma=\infty$, the proper scaling becomes $\sigma_{n}^{-1/2}$ and we obtain a convergence toward Aldous’s CRT.

Résumé

On s’intéresse à la limite d’échelle de grandes quadrangulations planaires à bord dont la longueur du bord est de l’ordre de la racine carrée du nombre de faces. On considère une suite $(\sigma_{n})$ d’entiers telle que $\sigma_{n}/\sqrt{2n}$ tende vers un certain $\sigma\in[0,\infty]$. Pour tout $n\ge1$, on note $\mathfrak{q}_{n}$ une carte aléatoire uniformément distribuée dans l’ensemble des quadrangulations planaires enracinées à bord ayant $n$ faces internes et $2\sigma_{n}$ demi-arêtes sur le bord. Dans le cas où $\sigma\in(0,\infty)$, on voit $\mathfrak{q}_{n}$ comme un espace métrique en munissant l’ensemble de ses sommets de la distance de graphe, renormalisée par le facteur $n^{-1/4}$. On montre que cet espace métrique converge en loi, tout du moins le long d’une sous-suite, vers un espace métrique limite aléatoire, au sens de la topologie de Gromov–Hausdorff. On montre que l’espace métrique limite est presque sûrement un espace de dimension de Hausdorff $4$ ayant un bord de dimension $2$ qui est homéomorphe au disque de dimension $2$. Pour $\sigma=0$, on a également la même convergence mais cette fois-ci, l’extraction d’une sous-suite n’est plus nécessaire et la limite est l’espace métrique connu sous le nom de carte brownienne. Pour $\sigma=\infty$, le bon facteur d’échelle devient $\sigma_{n}^{-1/2}$ et on a convergence vers l’arbre continu brownien d’Aldous.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 51, Number 2 (2015), 432-477.

Dates
First available in Project Euclid: 10 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1428672676

Digital Object Identifier
doi:10.1214/13-AIHP581

Mathematical Reviews number (MathSciNet)
MR3335010

Zentralblatt MATH identifier
1319.60067

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 57N05: Topology of $E^2$ , 2-manifolds 60C05: Combinatorial probability

Keywords
Random maps Random trees Brownian snake Scaling limits Regular convergence Gromov topology Hausdorff dimension Brownian CRT Random metric spaces

Citation

Bettinelli, Jérémie. Scaling limit of random planar quadrangulations with a boundary. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 2, 432--477. doi:10.1214/13-AIHP581. https://projecteuclid.org/euclid.aihp/1428672676


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