SMF - Publications - Annales scientifiques de l'ENS - Parutions

Résumé :
Mosaïques sur des cartes aléatoires en genre arbitraire
Nous examinons les propriétés de mosaïques de type Voronoï sur des quadrangulations bipartites de genre arbitraire. Ceci est rendu possible par une généralisation naturelle d'une bijection de Marcus et Schaeffer, permettant de décrire ces mosaïques par des cartes étiquetées avec un nombre fixé de faces, dont nous déterminons les limites d'échelle. Parmi les applications de ces résultats, figurent le comptage asymptotique des quadrangulations, ainsi que des propriétés métriques typiques de quadrangulations choisies au hasard. En particulier, nous montrons que les limites d'échelles de ces quadrangulations aléatoires sont telles que presque toute paire de points est liée par un unique chemin géodésique.

Mots-clefs : Cartes aléatoires, limites d'échelle, serpents aléatoires, comptage asymptotique, géodésiques

Abstract:
We investigate Voronoi-like tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing one to encode such structures by labeled maps with a fixed number of faces. We investigate the scaling limits of the latter. Applications include asymptotic enumeration results for quadrangulations, and typical metric properties of randomly sampled quadrangulations. In particular, we show that scaling limits of these random quadrangulations are such that almost every pair of points is linked by a unique geodesic.

Keywords: Random maps, scaling limits, random snakes, asymptotic enumeration, geodesics

Class. math. : 60C05, 05C30, 60F05

ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

Bibliographie:

1: Aldous, David; The continuum random tree. I; Ann. Probab. 19 (1991) 1–28; Math Reviews MR1085326; Zentralblatt 722.60013
2: Ambjørn, Jan and Durhuus, Bergfinnur and Jonsson, Thordur; Quantum geometry. A statistical field theory approach; Cambridge Univ. Press, 1997; Math Reviews MR1465433; Zentralblatt 993.82500
3: Angel, O.; Growth and percolation on the uniform infinite planar triangulation; Geom. Funct. Anal. 13 (2003) 935–974; Math Reviews MR2024412
4: Bender, Edward A. and Canfield, E. Rodney; The asymptotic number of rooted maps on a surface; J. Combin. Theory 43 (1986) 244–257; Math Reviews MR867650; Zentralblatt 606.05031
5: Billingsley, Patrick; Convergence of probability measures; John Wiley Sons Inc., 1999; Math Reviews MR1700749; Zentralblatt 944.60003
6: Bouttier, J. and Di Francesco, P. and Guitter, E.; Planar maps as labeled mobiles; Electron. J. Combin. 11 (2004) Research Paper 69 (electronic); Math Reviews MR2097335; Zentralblatt 1060.05045
7: Bouttier, J. and Guitter, E.; Statistics in geodesics in large quadrangulations; J. Phys. 41 (2008) 145001, 30; Math Reviews MR2450375
8: Bourbaki, Nicolas; Éléments de mathématique, Topologie générale, chap. 1 à 4; Hermann, 1971
9: Burago, Dmitri and Burago, Yuri and Ivanov, Sergei; A course in metric geometry; Amer. Math. Soc., 2001; Math Reviews MR1835418
10: Chapuy, G. and Marcus, M. and Schaeffer, Gilles; A bijection for rooted maps on orientable surfaces; (2007) arXiv:0712.3649
11: Chassaing, Philippe and Schaeffer, Gilles; Random planar lattices and integrated superBrownian excursion; Probab. Theory Related Fields 128 (2004) 161–212; Math Reviews MR2031225; Zentralblatt 1041.60008
12: Dudley, R. M.; Real analysis and probability; Cambridge Univ. Press, 2002; Math Reviews MR1932358; Zentralblatt 1023.60001
13: Duquesne, Thomas; A limit theorem for the contour process of conditioned Galton-Watson trees; Ann. Probab. 31 (2003) 996–1027; Math Reviews MR1964956; Zentralblatt 1025.60017
14: Duquesne, Thomas and Le Gall, Jean-François; Random trees, Lévy processes and spatial branching processes; Astérisque 281 (2002); Math Reviews MR1954248; Zentralblatt 1037.60074
15: Duquesne, Thomas and Le Gall, Jean-François; Probabilistic and fractal aspects of Lévy trees; Probab. Theory Related Fields 131 (2005) 553–603; Math Reviews MR2147221; Zentralblatt 1070.60076
16: Evans, Steven N. and Pitman, Jim and Winter, Anita; Rayleigh processes, real trees, and root growth with re-grafting; Probab. Theory Related Fields 134 (2006) 81–126; Math Reviews MR2221786; Zentralblatt 1086.60050
17: Evans, Steven N. and Winter, Anita; Subtree prune and regraft: a reversible real tree-valued Markov process; Ann. Probab. 34 (2006) 918–961; Math Reviews MR2243874; Zentralblatt 1101.60054
18: Flajolet, Philippe and Sedgewick, Robert; Analytic combinatorics; Cambridge Univ. Press, 2009; Math Reviews MR2483235; Zentralblatt pre05485323
19: Fukaya, Kenji; Collapsing of Riemannian manifolds and eigenvalues of Laplace operator; Invent. Math. 87 (1987) 517–547; Math Reviews MR874035; Zentralblatt 589.58034
20: Gao, Zhicheng; A pattern for the asymptotic number of rooted maps on surfaces; J. Combin. Theory 64 (1993) 246–264; Math Reviews MR1245161; Zentralblatt 792.05073
21: Greven, A. and Pfaffelhuber, P. and Winter, Anita; Convergence in distribution of random metric measure spaces (-coalescent measure trees); (2006)
22: Gromov, Misha; Metric structures for Riemannian and non-Riemannian spaces; Birkhäuser, 1999; Math Reviews MR1699320; Zentralblatt 953.53002
23: Janson, Svante and Marckert, Jean-François; Convergence of discrete snakes; J. Theoret. Probab. 18 (2005) 615–647; Math Reviews MR2167644; Zentralblatt 1084.60049
24: Lando, Sergei K. and Zvonkin, Alexander K.; Graphs on surfaces and their applications; Springer, 2004; Math Reviews MR2036721
25: Le Gall, Jean-François; The uniform random tree in a Brownian excursion; Probab. Theory Related Fields 96 (1993) 369–383; Math Reviews MR1231930; Zentralblatt 794.60080
26: Le Gall, Jean-François; The topological structure of scaling limits of large planar maps; Invent. Math. 169 (2007) 621–670; Math Reviews MR2336042; Zentralblatt 1132.60013
27: Le Gall, Jean-François; Geodesics in large planar maps and in the Brownian map; (2008) arXiv:0804.3012
28: Le Gall, Jean-François and Paulin, Frédéric; Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere; Geom. Funct. Anal. 18 (2008) 893–918; Math Reviews MR2438999; Zentralblatt pre05565912
29: Lévy, Thierry; Yang-Mills measure on compact surfaces; Mem. Amer. Math. Soc. 166 (2003); Math Reviews MR2006374
30: Marckert, Jean-François and Miermont, Grégory; Invariance principles for random bipartite planar maps; Ann. Probab. 35 (2007) 1642–1705; Math Reviews MR2349571; Zentralblatt pre05201525
31: Marckert, Jean-François and Mokkadem, Abdelkader; States spaces of the snake and its tour–convergence of the discrete snake; J. Theoret. Probab. 16 (2003) 1015–1046; Math Reviews MR2033196; Zentralblatt 1044.60083
32: Marckert, Jean-François and Mokkadem, Abdelkader; Limit of normalized quadrangulations: the Brownian map; Ann. Probab. 34 (2006) 2144–2202; Math Reviews MR2294979; Zentralblatt 1117.60038
33: Marcus, M. and Schaeffer, Gilles; Une bijection simple pour les cartes orientables; (2001) http://www.lix.polytechnique.fr/~schaeffe/Biblio/MaSc01.ps
34: Miermont, Grégory; An invariance principle for random planar maps; in Fourth Colloquium on Mathematics and Computer Sciences CMCS'06; Discrete Math. Theor. Comput. Sci. Proc. (2006) 39–58 (electronic)
35: Miermont, Grégory; Invariance principles for spatial multitype Galton-Watson trees; Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008) 1128–1161; Math Reviews MR2469338
36: Miermont, Grégory and Weill, Mathilde; Radius and profile of random planar maps with faces of arbitrary degrees; Electron. J. Probab. 13 (2008) 79–106; Math Reviews MR2375600
37: Okounkov, Andrei; Random matrices and random permutations; Int. Math. Res. Not. 2000 (2000) 1043–1095; Math Reviews MR1802530; Zentralblatt 1018.15020
38: Okounkov, Andrei and Pandharipande, R.; Gromov-Witten theory, Hurwitz numbers, and matrix models; in Algebraic geometry–Seattle 2005. Part 1; Proc. Sympos. Pure Math. 80 (2009) 325–414; Math Reviews MR2483941; Zentralblatt pre05548155
39: Petrov, V. V.; Sums of independent random variables; Springer, 1975; Math Reviews MR0388499; Zentralblatt 322.60043
40: Pitman, Jim; Combinatorial stochastic processes; Springer, 2006; Math Reviews MR2245368
41: Revuz, Daniel and Yor, Marc; Continuous martingales and Brownian motion; Springer, 1999; Math Reviews MR1725357; Zentralblatt 917.60006
42: Schaeffer, Gilles; Conjugaison d'arbres et cartes combinatoires aléatoires; Thèse de doctorat, Université Bordeaux I (1998)
43: Sheffield, Scott; Gaussian free fields for mathematicians; Probab. Theory Related Fields 139 (2007) 521–541; Math Reviews MR2322706; Zentralblatt 1132.60072
44: Villani, Cédric; Optimal transport; Springer, 2009; Math Reviews MR2459454; Zentralblatt 1156.53003
45: Weill, Mathilde; Asymptotics for rooted bipartite planar maps and scaling limits of two-type spatial trees; Electron. J. Probab. 12 (2007) 887–925; Math Reviews MR2318414; Zentralblatt 1127.05096