Abstract
We discuss scaling limits of large bipartite planar maps. If p≥2 is a fixed integer, we consider, for every integer n≥2, a random planar map M n which is uniformly distributed over the set of all rooted 2p-angulations with n faces. Then, at least along a suitable subsequence, the metric space consisting of the set of vertices of M n , equipped with the graph distance rescaled by the factor n -1/4, converges in distribution as n→∞ towards a limiting random compact metric space, in the sense of the Gromov–Hausdorff distance. We prove that the topology of the limiting space is uniquely determined independently of p and of the subsequence, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the limit is almost surely equal to 4.
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Abraham, R., Werner, W.: Avoiding probabilities for Brownian snakes and super-Brownian motion. Electron. J. Probab. 2(3), 27 (1997)
Aldous, D.: The continuum random tree I. Ann. Probab. 19, 1–28 (1991)
Aldous, D.: The continuum random tree III. Ann. Probab. 21, 248–289 (1993)
Ambjorn, J., Durhuus, B., Jonsson, T.: Quantum Geometry. A Statistical Field Theory Approach. Cambr. Monogr. Math. Phys., vol. 1. Cambridge Univ. Press, Cambridge (1997)
Angel, O.: Growth and percolation on the uniform infinite planar triangulation. Geom. Funct. Anal. 3, 935–974 (2003)
Angel, O., Schramm, O.: Uniform infinite planar triangulations. Commun. Math. Phys. 241, 191–213 (2003)
Bouttier, J.: Physique statistique des surfaces aléatoires et combinatoire bijective des cartes planaires. PhD thesis, Université Paris 6. (2005) http://tel.ccsd.cnrs.fr/documents/archives0/00/01/06/51/index.html
Bouttier, J., Di Francesco, P., Guitter, E.: Planar maps as labeled mobiles. Electron. J. Comb. 11, #R69. (2004)
Brézin, E., Itzykson, C., Parisi, G., Zuber, J.B.: Planar diagrams. Commun. Math. Phys. 59, 35–51 (1978)
Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. Grad. Stud. Math., vol. 33. AMS, Boston (2001)
Chassaing, P., Durhuus, B.: Local limit of labeled trees and expected volume growth in a random quadrangulation. Ann. Probab. 34, 879–917 (2006)
Chassaing, P., Schaeffer, G.: Random planar lattices and integrated super Brownian excursion. Probab. Theory Relat. Fields 128, 161–212 (2004)
Cori, R., Vauquelin, B.: Planar maps are well labeled trees. Can. J. Math. 33, 1023–1042 (1981)
David, F.: Planar diagrams, two-dimensional lattice gravity and surface models. Nucl. Phys. B 257, 45–58 (1985)
Duquesne, T., Le Gall, J.F.: Probabilistic and fractal aspects of Lévy trees. Probab. Theory Relat. Fields 131, 553–603 (2005)
Ethier, S.N., Kurtz, T.: Markov Processes: Characterization and Convergence. Wiley, New York (1986)
Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhäuser, Boston (2001)
’t Hooft, G.: A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461–473 (1974)
Janson, S., Marckert, J.F.: Convergence of discrete snakes. J. Theor. Probab. 18, 615–645 (2005)
Krikun, M.: Local structure of random quadrangulations. Preprint (2005) arxiv:math.PR/0512304
Le Gall, J.F.: Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lect. Math. ETH Zürich. Birkhäuser, Boston (1999)
Le Gall, J.F.: Random trees and applications. Probab. Surv. 2, 245–311 (2005)
Le Gall, J.F.: A conditional limit theorem for tree-indexed random walk. Stochastic Processes Appl. 116, 539–567 (2006)
Le Gall, J.F., Paulin, F.: Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere. Preprint (2006) arXiv:math.PR/0612315
Le Gall, J.F., Weill, M.: Conditioned Brownian trees. Ann. Inst. Henri Poincaré, Probab. Stat. 42, 455–489 (2006)
Marckert, J.F., Miermont, G.: Invariance principles for labeled mobiles and bipartite planar maps. To appear in Ann. Probab. (2005) arXiv:math.PR/0504110
Marckert, J.F., Mokkadem, A.: Limit of normalized quadrangulations. The Brownian map. Ann. Probab. 34, 2144–2202 (2006)
Neveu, J.: Arbres et processus de Galton-Watson. Ann. Inst. Henri Poincaré, Probab. Stat. 22, 199–207 (1986)
Schaeffer, G.: Conjugaison d’arbres et cartes combinatoires aléatoires. PhD thesis, Université Bordeaux I. (1998) http://www.lix.polytechnique.fr/∼schaeffe/Biblio/
Tutte, W.T.: A census of planar maps. Can. J. Math. 15, 249–271 (1963)
Weill, M.: Asymptotics for rooted planar maps and scaling limits of two-type Galton-Watson trees. To appear in Electron. J. Probab. (2007) arXiv:math.PR/0609334
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Le Gall, JF. The topological structure of scaling limits of large planar maps. Invent. math. 169, 621–670 (2007). https://doi.org/10.1007/s00222-007-0059-9
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DOI: https://doi.org/10.1007/s00222-007-0059-9