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The topological structure of scaling limits of large planar maps. (English) Zbl 1132.60013
The purpose of this work is to investigate continuous limits of rescaled planar maps. Let \(p\) be a fixed integer \((p\geq2)\) and consider for every integer \(n\geq2\) 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 \(n\to\infty\) towards a limiting random compact metric space, in the sense of the Gromov-Hausdorff distance. The author proves 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.
Finally, the last section contains the calculation of the Hausdorff dimension of the limiting metric space.

60E05 Probability distributions: general theory
60B05 Probability measures on topological spaces
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[1] Abraham, R., Werner, W.: Avoiding probabilities for Brownian snakes and super-Brownian motion. Electron. J. Probab. 2(3), 27 (1997) · Zbl 0890.60068
[2] Aldous, D.: The continuum random tree I. Ann. Probab. 19, 1–28 (1991) · Zbl 0722.60013
[3] Aldous, D.: The continuum random tree III. Ann. Probab. 21, 248–289 (1993) · Zbl 0791.60009
[4] Ambjorn, J., Durhuus, B., Jonsson, T.: Quantum Geometry. A Statistical Field Theory Approach. Cambr. Monogr. Math. Phys., vol. 1. Cambridge Univ. Press, Cambridge (1997) · Zbl 0993.82500
[5] Angel, O.: Growth and percolation on the uniform infinite planar triangulation. Geom. Funct. Anal. 3, 935–974 (2003) · Zbl 1039.60085
[6] Angel, O., Schramm, O.: Uniform infinite planar triangulations. Commun. Math. Phys. 241, 191–213 (2003) · Zbl 1098.60010
[7] 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
[8] Bouttier, J., Di Francesco, P., Guitter, E.: Planar maps as labeled mobiles. Electron. J. Comb. 11, #R69. (2004) · Zbl 1060.05045
[9] Brézin, E., Itzykson, C., Parisi, G., Zuber, J.B.: Planar diagrams. Commun. Math. Phys. 59, 35–51 (1978) · Zbl 0997.81548
[10] Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. Grad. Stud. Math., vol. 33. AMS, Boston (2001) · Zbl 0981.51016
[11] Chassaing, P., Durhuus, B.: Local limit of labeled trees and expected volume growth in a random quadrangulation. Ann. Probab. 34, 879–917 (2006) · Zbl 1102.60007
[12] Chassaing, P., Schaeffer, G.: Random planar lattices and integrated super Brownian excursion. Probab. Theory Relat. Fields 128, 161–212 (2004) · Zbl 1041.60008
[13] Cori, R., Vauquelin, B.: Planar maps are well labeled trees. Can. J. Math. 33, 1023–1042 (1981) · Zbl 0468.05024
[14] David, F.: Planar diagrams, two-dimensional lattice gravity and surface models. Nucl. Phys. B 257, 45–58 (1985)
[15] Duquesne, T., Le Gall, J.F.: Probabilistic and fractal aspects of Lévy trees. Probab. Theory Relat. Fields 131, 553–603 (2005) · Zbl 1070.60076
[16] Ethier, S.N., Kurtz, T.: Markov Processes: Characterization and Convergence. Wiley, New York (1986) · Zbl 0592.60049
[17] Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhäuser, Boston (2001)
[18] ’t Hooft, G.: A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461–473 (1974)
[19] Janson, S., Marckert, J.F.: Convergence of discrete snakes. J. Theor. Probab. 18, 615–645 (2005) · Zbl 1084.60049
[20] Krikun, M.: Local structure of random quadrangulations. Preprint (2005) arxiv:math.PR/0512304
[21] Le Gall, J.F.: Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lect. Math. ETH Zürich. Birkhäuser, Boston (1999) · Zbl 0938.60003
[22] Le Gall, J.F.: Random trees and applications. Probab. Surv. 2, 245–311 (2005) · Zbl 1189.60161
[23] Le Gall, J.F.: A conditional limit theorem for tree-indexed random walk. Stochastic Processes Appl. 116, 539–567 (2006) · Zbl 1093.60061
[24] Le Gall, J.F., Paulin, F.: Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere. Preprint (2006) arXiv:math.PR/0612315 · Zbl 1166.60006
[25] Le Gall, J.F., Weill, M.: Conditioned Brownian trees. Ann. Inst. Henri Poincaré, Probab. Stat. 42, 455–489 (2006) · Zbl 1107.60053
[26] Marckert, J.F., Miermont, G.: Invariance principles for labeled mobiles and bipartite planar maps. To appear in Ann. Probab. (2005) arXiv:math.PR/0504110
[27] Marckert, J.F., Mokkadem, A.: Limit of normalized quadrangulations. The Brownian map. Ann. Probab. 34, 2144–2202 (2006) · Zbl 1117.60038
[28] Neveu, J.: Arbres et processus de Galton-Watson. Ann. Inst. Henri Poincaré, Probab. Stat. 22, 199–207 (1986) · Zbl 0601.60082
[29] Schaeffer, G.: Conjugaison d’arbres et cartes combinatoires aléatoires. PhD thesis, Université Bordeaux I. (1998) http://www.lix.polytechnique.fr/chaeffe/Biblio/
[30] Tutte, W.T.: A census of planar maps. Can. J. Math. 15, 249–271 (1963) · Zbl 0115.17305
[31] 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 · Zbl 1127.05096
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