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.
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.
Mots clés : 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, Tome 51 (2015) no. 2, pp. 432-477. doi : 10.1214/13-AIHP581. http://www.numdam.org/articles/10.1214/13-AIHP581/
[1] The continuum random tree. I. Ann. Probab. 19 (1) (1991) 1–28. | MR | Zbl
.[2] The continuum random tree. III. Ann. Probab. 21 (1) (1993) 248–289. | MR | Zbl
.[3] Regular convergence. Duke Math. J. 11 (1944) 441–450. | DOI | MR | Zbl
.[4] The number of degree-restricted rooted maps on the sphere. SIAM J. Discrete Math. 7 (1) (1994) 9–15. | MR | Zbl
and .[5] Increase of a Lévy process with no positive jumps. Stochastics Stochastics Rep. 37 (4) (1991) 247–251. | MR | Zbl
.[6] Path transformations of first passage bridges. Electron. Commun. Probab. 8 (2003) 155–166 (electronic). | DOI | MR | Zbl
, and .[7] Scaling limits for random quadrangulations of positive genus. Electron. J. Probab. 15 (52) (2010) 1594–1644. | MR | Zbl
.[8] The topology of scaling limits of positive genus random quadrangulations. Ann. Probab. 40 (5) (2012) 1897–1944. | MR | Zbl
.[9] Convergence of Probability Measures. Wiley, New York, 1968. | MR | Zbl
.[10] A recursive approach to the model on random maps via nested loops. J. Phys. A 45 (2012) 045002. | MR | Zbl
, and .[11] Planar maps as labeled mobiles. Electron. J. Combin. 11 (1) (2004) Research Paper 69 (electronic). | DOI | MR | Zbl
, and .[12] Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop. J. Phys. A 42 (46) (2009) 465208. | DOI | MR | Zbl
and .[13] A Course in Metric Geometry. Graduate Studies in Mathematics 33. American Mathematical Society, Providence, RI, 2001. | DOI | MR | Zbl
, and .[14] A bijection for rooted maps on orientable surfaces. SIAM J. Discrete Math. 23 (3) (2009) 1587–1611. | MR | Zbl
, and .[15] Random planar lattices and integrated super-Brownian excursion. Probab. Theory Related Fields 128 (2) (2004) 161–212. | MR | Zbl
and .[16] Planar maps are well labeled trees. Canad. J. Math. 33 (5) (1981) 1023–1042. | MR | Zbl
and .[17] The Brownian cactus I. Scaling limits of discrete cactuses. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013) 340–373. | Numdam | MR | Zbl
, and .[18] Uniform infinite planar quadrangulations with a boundary. Random Structures Algorithms. To appear, 2015. Available at arXiv:1202.5452. | DOI | MR | Zbl
and .[19] Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002) vi147. | Numdam | MR | Zbl
and .[20] Geometric Measure Theory. Die Grundlehren der Mathematischen Wissenschaften 153. Springer, New York, 1969. | MR | Zbl
.[21] Convergence in distribution of random metric measure spaces1–2) (2009) 285–322. | MR | Zbl
, and .[22] Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics 152. Birkhäuser, Boston, MA, 1999. Based on the 1981 French original [MR0682063], with appendices by M. Katz, P. Pansu and S. Semmes, translated from the French by Sean Michael Bates. | MR | Zbl
.[23] The smallest uniform upper bound on the distance between the mean and the median of the binomial and Poisson distributions. Statist. Probab. Lett. 23 (1) (1995) 21–25. | MR | Zbl
.[24] Poisson Processes. Oxford Studies in Probability 3. Oxford Univ. Press, New York, 1993. | MR | Zbl
.[25] Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, 1999. | MR | Zbl
.[26] Random trees and applications. Probab. Surv. 2 (2005) 245–311 (electronic). | MR | Zbl
.[27] The topological structure of scaling limits of large planar maps. Invent. Math. 169 (3) (2007) 621–670. | MR | Zbl
.[28] Geodesics in large planar maps and in the Brownian map. Acta Math. 205 (2) (2010) 287–360. | MR | Zbl
.[29] Uniqueness and universality of the Brownian map. Ann. Probab. 41 (2013) 2880–2960. | MR | Zbl
.[30] Scaling limits of random planar maps with large faces. Ann. Probab. 39 (1) (2011) 1–69. | MR | Zbl
and .[31] Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere. Geom. Funct. Anal. 18 (3) (2008) 893–918. | MR | Zbl
and .[32] Conditioned Brownian trees. Ann. Inst. Henri Poincaré Probab. Stat. 42 (4) (2006) 455–489. | Numdam | MR | Zbl
and .[33] Limit of normalized quadrangulations: The Brownian map. Ann. Probab. 34 (6) (2006) 2144–2202. | MR | Zbl
and .[34] On the sphericity of scaling limits of random planar quadrangulations. Electron. Commun. Probab. 13 (2008) 248–257. | DOI | MR | Zbl
.[35] Tessellations of random maps of arbitrary genus. Ann. Sci. Éc. Norm. Supér. (4) 42 (5) (2009) 725–781. | Numdam | MR | Zbl
.[36] The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210 (2013) 319–401. | DOI | MR | Zbl
.[37] Limit Theorems of Probability Theory Sequences of Independent Random Variables. Oxford Studies in Probability 4. Oxford Univ. Press, New York, 1995. | MR | Zbl
.[38] Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin, 1999. | DOI | MR | Zbl
and .[39] Bijective census and random generation of Eulerian planar maps with prescribed vertex degrees. Electron. J. Combin. 4 (1) (1997) Research Paper 20 (electronic). | DOI | MR | Zbl
.[40] Conjugaison d’arbres et cartes combinatoires aléatoires. Ph.D. thesis, Univ. Bordeaux 1, 1998.
.[41] Covering the line with random intervals. Z. Wahrsch. Verw. Gebiete 23 (1972) 163–170. | DOI | MR | Zbl
.[42] A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 (1) (1979) 143–149. | MR | Zbl
.[43] On sequences and limiting sets. Fund. Math. 25 (1935) 408–426. | DOI | JFM
.[44] Regular convergence and monotone transformations. Amer. J. Math. 57 (4) (1935) 902–906. | JFM | MR
.Cité par Sources :