Scaling limit of random planar quadrangulations with a boundary
Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 2, p. 432-477

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 /2n tends to some σ[0,]. For every n1, 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 2σ n half-edges on the boundary. For σ(0,), 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=, the proper scaling becomes σ 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 (σ n ) d’entiers telle que σ n /2n tende vers un certain σ[0,]. Pour tout n1, on note 𝔮 n une carte aléatoire uniformément distribuée dans l’ensemble des quadrangulations planaires enracinées à bord ayant n faces internes et 2σ n demi-arêtes sur le bord. Dans le cas où σ(0,), 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=, le bon facteur d’échelle devient σ n -1/2 et on a convergence vers l’arbre continu brownien d’Aldous.

DOI : https://doi.org/10.1214/13-AIHP581
Classification:  60F17,  60D05,  57N05,  60C05
Keywords: random maps, random trees, brownian snake, scaling limits, regular convergence, Gromov topology, Hausdorff dimension, brownian CRT, random metric spaces
@article{AIHPB_2015__51_2_432_0,
     author = {Bettinelli, J\'er\'emie},
     title = {Scaling limit of random planar quadrangulations with a boundary},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {2},
     year = {2015},
     pages = {432-477},
     doi = {10.1214/13-AIHP581},
     zbl = {1319.60067},
     mrnumber = {3335010},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2015__51_2_432_0}
}
Bettinelli, Jérémie. Scaling limit of random planar quadrangulations with a boundary. Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 2, pp. 432-477. doi : 10.1214/13-AIHP581. http://www.numdam.org/item/AIHPB_2015__51_2_432_0/

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