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 Electronic Journal of Probability > Vol. 15(2010) > Paper 52 open journal systems 


Scaling Limits for Random Quadrangulations of Positive Genus

Jérémie L Bettinelli, Université Paris Sud


Abstract
Abstract. We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given g, we consider, for every positive integer n, a random quadrangulation q_n uniformly distributed over the set of all rooted bipartite quadrangulations of genus g with n faces. We view it as a metric space by endowing its set of vertices with the graph distance. We show that, as n tends to infinity, this metric space, with distances rescaled by the factor n to the power of -1/4, converges in distribution, at least along some subsequence, toward a limiting random metric space. This convergence holds in the sense of the Gromov-Hausdorff topology on compact metric spaces. We show that, regardless of the choice of the subsequence, the Hausdorff dimension of the limiting space is almost surely equal to 4. Our main tool is a bijection introduced by Chapuy, Marcus, and Schaeffer between the quadrangulations we consider and objects they call well-labeled g-trees. An important part of our study consists in determining the scaling limits of the latter.


Full text: PDF

Pages: 1594-1644

Published on: October 20, 2010
Bibliography
  1. Bender, Edward A.; Canfield, E. Rodney. The number of rooted maps on an orientable surface. J. Combin. Theory Ser. B 53 (1991), no. 2, 293--299. MR1129556 (92g:05100)
  2. Bertoin, Jean; Chaumont, Loïc; Pitman, Jim. Path transformations of first passage bridges. Electron. Comm. Probab. 8 (2003), 155--166 (electronic). MR2042754 (2005d:60132)
  3. Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp. MR0233396 (38 #1718)
  4. Burago, Dmitri; Burago, Yuri; Ivanov, Sergei. A course in metric geometry. Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001. xiv+415 pp. ISBN: 0-8218-2129-6 MR1835418 (2002e:53053)
  5. Chapuy, Guillaume. The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees. Probability Theory and Related Fields 147 (2010), no. 3, 415--447. Math. Review number not available
  6. Chapuy, Guillaume; Marcus, Michel; Schaeffer, Gilles. A bijection for rooted maps on orientable surfaces. SIAM J. Discrete Math. 23 (2009), no. 3, 1587--1611. MR2563085 (2010k:05142)
  7. Chassaing, Philippe; Schaeffer, Gilles. Random planar lattices and integrated superBrownian excursion. Probab. Theory Related Fields 128 (2004), no. 2, 161--212. MR2031225 (2004k:60016)
  8. Cori, Robert; Vauquelin, Bernard. Planar maps are well labeled trees. Canad. J. Math. 33 (1981), no. 5, 1023--1042. MR0638363 (83c:05070)
  9. Duquesne, Thomas; Le Gall, Jean-François. Random trees, Lévy processes and spatial branching processes. Astérisque No. 281 (2002), vi+147 pp. MR1954248 (2003m:60239)
  10. Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8 MR0838085 (88a:60130)
  11. Federer, Herbert. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969 xiv+676 pp. MR0257325 (41 #1976)
  12. Fitzsimmons, Pat; Pitman, Jim; Yor, Marc. Markovian bridges: construction, Palm interpretation, and splicing. Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992), 101--134, Progr. Probab., 33, Birkhäuser Boston, Boston, MA, 1993. MR1278079 (95i:60070)
  13. Gao, Zhicheng. The number of degree restricted maps on general surfaces. Discrete Math. 123 (1993), no. 1-3, 47--63. MR1256081 (94j:05008)
  14. Gromov, Misha. Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original [MR0682063 (85e:53051)]. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkhäuser Boston, Inc., Boston, MA, 1999. xx+585 pp. ISBN: 0-8176-3898-9 MR1699320 (2000d:53065)
  15. Le Gall, Jean-François. Master course: Mouvement brownien, processus de branchement et superprocessus, 1994. Available on http://www.dma.ens.fr/~legall/
  16. Le Gall, Jean-François. Spatial branching processes, random snakes and partial differential equations. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1999. x+163 pp. ISBN: 3-7643-6126-3 MR1714707 (2001g:60211)
  17. Le Gall, Jean-François. The topological structure of scaling limits of large planar maps. Invent. Math. 169 (2007), no. 3, 621--670. MR2336042 (2008i:60022)
  18. Le Gall, Jean-François; Miermont, Grégory. Scaling limits of random planar maps with large faces. arXiv:0907.3262 to appear in Ann. Probab. 2009.
  19. Le Gall, Jean-François; Paulin, Frédéric. Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere. Geom. Funct. Anal. 18 (2008), no. 3, 893--918. MR2438999 (2010a:60030)
  20. Marckert, Jean-François; Mokkadem, Abdelkader. States spaces of the snake and its tour---convergence of the discrete snake. J. Theoret. Probab. 16 (2003), no. 4, 1015--1046 (2004). MR2033196 (2005c:60095)
  21. Marckert, Jean-François; Mokkadem, Abdelkader. Limit of normalized quadrangulations: the Brownian map. Ann. Probab. 34 (2006), no. 6, 2144--2202. MR2294979 (2007m:60092)
  22. Miermont, Grégory. On the sphericity of scaling limits of random planar quadrangulations. Electron. Commun. Probab. 13 (2008), 248--257. MR2399286 (2009d:60024)
  23. Miermont, Grégory. Tessellations of random maps of arbitrary genus. Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 5, 725--781. MR2571957
  24. Neveu, J. Arbres et processus de Galtonmhy Watson. (French) [Galton-Watson trees and processes] Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 2, 199--207. MR0850756 (88a:60150)
  25. Petrov, V. V. Sums of independent random variables. Translated from the Russian by A. A. Brown. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. Springer-Verlag, New York-Heidelberg, 1975. x+346 pp. MR0388499 (52 #9335)
  26. Petrov, Valentin V. Limit theorems of probability theory. Sequences of independent random variables. Oxford Studies in Probability, 4. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. xii+292 pp. ISBN: 0-19-853499-X MR1353441 (96h:60048)
  27. Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 MR1725357 (2000h:60050)
  28. Schaeffer, Gilles. Conjugaison d'arbres et cartes combinatoires aléatoires. PhD thesis, Université de Bordeaux 1, 1998.
  29. Schwartz, Laurent. Analyse. (French) Deuxième partie: Topologie générale et analyse fonctionnelle. Collection Enseignement des Sciences, No. 11. Hermann, Paris, 1970. 433 pp. MR0467223 (57 #7087)
  30. Stroock, Daniel W. Probability theory, an analytic view. Cambridge University Press, Cambridge, 1993. xvi+512 pp. ISBN: 0-521-43123-9 MR1267569 (95f:60003)
















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Electronic Journal of Probability. ISSN: 1083-6489