Scaling limit of random planar quadrangulations with a boundary

Bettinelli, Jérémie

doi:10.1214/13-AIHP581

Bettinelli, Jérémie

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 432-477.

Résumé
Abstract

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 $(σ_{n})$ d’entiers telle que $σ_{n} / \sqrt{2 n}$ tende vers un certain $σ \in [0, \infty]$ . Pour tout $n \geq 1$ , on note $𝔮_{n}$ une carte aléatoire uniformément distribuée dans l’ensemble des quadrangulations planaires enracinées à bord ayant $n$ faces internes et $2 σ_{n}$ demi-arêtes sur le bord. Dans le cas où $σ \in (0, \infty)$ , on voit $𝔮_{n}$ 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 $σ = 0$ , 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 $σ = \infty$ , le bon facteur d’échelle devient $σ_{n}^{- 1 / 2}$ et on a convergence vers l’arbre continu brownien d’Aldous.

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 $(σ_{n})$ of integers such that $σ_{n} / \sqrt{2 n}$ tends to some $σ \in [0, \infty]$ . For every $n \geq 1$ , we denote by $𝔮_{n}$ a random map uniformly distributed over the set of all rooted planar quadrangulations with a boundary having $n$ faces and $2 σ_{n}$ half-edges on the boundary. For $σ \in (0, \infty)$ , we view $𝔮_{n}$ 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 $σ = 0$ , the same convergence holds without extraction and the limit is the so-called Brownian map. For $σ = \infty$ , the proper scaling becomes $σ_{n}^{- 1 / 2}$ and we obtain a convergence toward Aldous’s CRT.

MR Zbl | 4 citations dans Numdam

DOI : 10.1214/13-AIHP581

Classification : 60F17, 60D05, 57N05, 60C05
Mots-clés : random maps, random trees, brownian snake, scaling limits, regular convergence, Gromov topology, Hausdorff dimension, brownian CRT, random metric spaces

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     title = {Scaling limit of random planar quadrangulations with a boundary},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {432--477},
     publisher = {Gauthier-Villars},
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     doi = {10.1214/13-AIHP581},
     mrnumber = {3335010},
     zbl = {1319.60067},
     language = {en},
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Bettinelli, Jérémie. Scaling limit of random planar quadrangulations with a boundary. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 432-477. doi : 10.1214/13-AIHP581. https://www.numdam.org/articles/10.1214/13-AIHP581/

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