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.

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.

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.

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
@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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1214/13-AIHP581/

[1] D. Aldous. The continuum random tree. I. Ann. Probab. 19 (1) (1991) 1–28. | MR | Zbl

[2] D. Aldous. The continuum random tree. III. Ann. Probab. 21 (1) (1993) 248–289. | MR | Zbl

[3] E. G. Begle. Regular convergence. Duke Math. J. 11 (1944) 441–450. | DOI | MR | Zbl

[4] E. A. Bender and E. R. Canfield. The number of degree-restricted rooted maps on the sphere. SIAM J. Discrete Math. 7 (1) (1994) 9–15. | MR | Zbl

[5] J. Bertoin. Increase of a Lévy process with no positive jumps. Stochastics Stochastics Rep. 37 (4) (1991) 247–251. | MR | Zbl

[6] J. Bertoin, L. Chaumont and J. Pitman. Path transformations of first passage bridges. Electron. Commun. Probab. 8 (2003) 155–166 (electronic). | DOI | MR | Zbl

[7] J. Bettinelli. Scaling limits for random quadrangulations of positive genus. Electron. J. Probab. 15 (52) (2010) 1594–1644. | MR | Zbl

[8] J. Bettinelli. The topology of scaling limits of positive genus random quadrangulations. Ann. Probab. 40 (5) (2012) 1897–1944. | MR | Zbl

[9] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968. | MR | Zbl

[10] G. Borot, J. Bouttier and E. Guitter. A recursive approach to the O(n) model on random maps via nested loops. J. Phys. A 45 (2012) 045002. | MR | Zbl

[11] J. Bouttier, P. Di Francesco and E. Guitter. Planar maps as labeled mobiles. Electron. J. Combin. 11 (1) (2004) Research Paper 69 (electronic). | DOI | MR | Zbl

[12] J. Bouttier and E. Guitter. Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop. J. Phys. A 42 (46) (2009) 465208. | DOI | MR | Zbl

[13] D. Burago, Y. Burago and S. Ivanov. A Course in Metric Geometry. Graduate Studies in Mathematics 33. American Mathematical Society, Providence, RI, 2001. | DOI | MR | Zbl

[14] G. Chapuy, M. Marcus and G. Schaeffer. A bijection for rooted maps on orientable surfaces. SIAM J. Discrete Math. 23 (3) (2009) 1587–1611. | MR | Zbl

[15] P. Chassaing and G. Schaeffer. Random planar lattices and integrated super-Brownian excursion. Probab. Theory Related Fields 128 (2) (2004) 161–212. | MR | Zbl

[16] R. Cori and B. Vauquelin. Planar maps are well labeled trees. Canad. J. Math. 33 (5) (1981) 1023–1042. | MR | Zbl

[17] N. Curien, J.-F. Le Gall and G. Miermont. The Brownian cactus I. Scaling limits of discrete cactuses. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013) 340–373. | Numdam | MR | Zbl

[18] N. Curien and G. Miermont. Uniform infinite planar quadrangulations with a boundary. Random Structures Algorithms. To appear, 2015. Available at arXiv:1202.5452. | DOI | MR | Zbl

[19] T. Duquesne and J.-F. Le Gall. Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002) vi+147. | Numdam | MR | Zbl

[20] H. Federer. Geometric Measure Theory. Die Grundlehren der Mathematischen Wissenschaften 153. Springer, New York, 1969. | MR | Zbl

[21] A. Greven, P. Pfaffelhuber and A. Winter. Convergence in distribution of random metric measure spaces1–2) (2009) 285–322. | MR | Zbl

[22] M. Gromov. 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] K. Hamza. 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] J. F. C. Kingman. Poisson Processes. Oxford Studies in Probability 3. Oxford Univ. Press, New York, 1993. | MR | Zbl

[25] J.-F. Le Gall. Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, 1999. | MR | Zbl

[26] J.-F. Le Gall. Random trees and applications. Probab. Surv. 2 (2005) 245–311 (electronic). | MR | Zbl

[27] J.-F. Le Gall. The topological structure of scaling limits of large planar maps. Invent. Math. 169 (3) (2007) 621–670. | MR | Zbl

[28] J.-F. Le Gall. Geodesics in large planar maps and in the Brownian map. Acta Math. 205 (2) (2010) 287–360. | MR | Zbl

[29] J.-F. Le Gall. Uniqueness and universality of the Brownian map. Ann. Probab. 41 (2013) 2880–2960. | MR | Zbl

[30] J.-F. Le Gall and G. Miermont. Scaling limits of random planar maps with large faces. Ann. Probab. 39 (1) (2011) 1–69. | MR | Zbl

[31] J.-F. Le Gall and F. Paulin. Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere. Geom. Funct. Anal. 18 (3) (2008) 893–918. | MR | Zbl

[32] J.-F. Le Gall and M. Weill. Conditioned Brownian trees. Ann. Inst. Henri Poincaré Probab. Stat. 42 (4) (2006) 455–489. | Numdam | MR | Zbl

[33] J.-F. Marckert and A. Mokkadem. Limit of normalized quadrangulations: The Brownian map. Ann. Probab. 34 (6) (2006) 2144–2202. | MR | Zbl

[34] G. Miermont. On the sphericity of scaling limits of random planar quadrangulations. Electron. Commun. Probab. 13 (2008) 248–257. | DOI | MR | Zbl

[35] G. Miermont. Tessellations of random maps of arbitrary genus. Ann. Sci. Éc. Norm. Supér. (4) 42 (5) (2009) 725–781. | Numdam | MR | Zbl

[36] G. Miermont. The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210 (2013) 319–401. | DOI | MR | Zbl

[37] V. V. Petrov. Limit Theorems of Probability Theory Sequences of Independent Random Variables. Oxford Studies in Probability 4. Oxford Univ. Press, New York, 1995. | MR | Zbl

[38] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin, 1999. | DOI | MR | Zbl

[39] G. Schaeffer. 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] G. Schaeffer. Conjugaison d’arbres et cartes combinatoires aléatoires. Ph.D. thesis, Univ. Bordeaux 1, 1998.

[41] L. A. Shepp. Covering the line with random intervals. Z. Wahrsch. Verw. Gebiete 23 (1972) 163–170. | DOI | MR | Zbl

[42] W. Vervaat. A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 (1) (1979) 143–149. | MR | Zbl

[43] G. T. Whyburn. On sequences and limiting sets. Fund. Math. 25 (1935) 408–426. | DOI | JFM

[44] G. T. Whyburn. Regular convergence and monotone transformations. Amer. J. Math. 57 (4) (1935) 902–906. | JFM | MR

Cité par Sources :