Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Scaling limit of random planar quadrangulations with a boundary

Jérémie Bettinelli

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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.


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

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

First available in Project Euclid: 10 April 2015

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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

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


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.

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