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 of integers such that tends to some . For every , we denote by a random map uniformly distributed over the set of all rooted planar quadrangulations with a boundary having faces and half-edges on the boundary. For , we view as a metric space by endowing its set of vertices with the graph metric, rescaled by . 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 with a boundary of Hausdorff dimension that is homeomorphic to the two-dimensional disc. For , the same convergence holds without extraction and the limit is the so-called Brownian map. For , the proper scaling becomes 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 d’entiers telle que tende vers un certain . Pour tout , on note une carte aléatoire uniformément distribuée dans l’ensemble des quadrangulations planaires enracinées à bord ayant faces internes et demi-arêtes sur le bord. Dans le cas où , on voit comme un espace métrique en munissant l’ensemble de ses sommets de la distance de graphe, renormalisée par le facteur . 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 ayant un bord de dimension qui est homéomorphe au disque de dimension . Pour , 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 , le bon facteur d’échelle devient et on a convergence vers l’arbre continu brownien d’Aldous.
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}, pages = {432--477}, publisher = {Gauthier-Villars}, volume = {51}, number = {2}, year = {2015}, doi = {10.1214/13-AIHP581}, mrnumber = {3335010}, zbl = {1319.60067}, language = {en}, url = {http://www.numdam.org/articles/10.1214/13-AIHP581/} }
TY - JOUR AU - Bettinelli, Jérémie TI - Scaling limit of random planar quadrangulations with a boundary JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2015 SP - 432 EP - 477 VL - 51 IS - 2 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/13-AIHP581/ DO - 10.1214/13-AIHP581 LA - en ID - AIHPB_2015__51_2_432_0 ER -
%0 Journal Article %A Bettinelli, Jérémie %T Scaling limit of random planar quadrangulations with a boundary %J Annales de l'I.H.P. Probabilités et statistiques %D 2015 %P 432-477 %V 51 %N 2 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/13-AIHP581/ %R 10.1214/13-AIHP581 %G en %F 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/articles/10.1214/13-AIHP581/
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