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The Brownian map is the scaling limit of uniform random plane quadrangulations

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Acta Mathematica

Abstract

We prove that uniform random quadrangulations of the sphere with n faces, endowed with the usual graph distance and renormalized by n −1/4, converge as n → ∞ in distribution for the Gromov–Hausdorff topology to a limiting metric space. We validate a conjecture by Le Gall, by showing that the limit is (up to a scale constant) the so-called Brownian map, which was introduced by Marckert–Mokkadem and Le Gall as the most natural candidate for the scaling limit of many models of random plane maps. The proof relies strongly on the concept of geodesic stars in the map, which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere.

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References

  1. Ambjørn, J., Durhuus, B. & Jonsson, T., Quantum Geometry. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1997.

  2. Bettinelli J.: Scaling limits for random quadrangulations of positive genus. Electron. J. Probab., 15(52), 1594–1644 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Bettinelli J.: The topology of scaling limits of positive genus random quadrangulations. Ann. Probab., 40(5), 1897–1944 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bouttier, J. & Guitter, E., The three-point function of planar quadrangulations. J. Stat. Mech. Theory Exp., 2008:7 (2008), P07020, 39 pp.

  5. Burago, D., Burago, Y. & Ivanov, S., A Course in Metric Geometry. Graduate Studies in Mathematics, 33. Amer. Math. Soc., Providence, RI, 2001.

  6. Chapuy G., Marcus M., Schaeffer G.: A bijection for rooted maps on orientable surfaces. SIAM J. Discrete Math., 23(3), 1587–1611 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chassaing P., Schaeffer G.: Random planar lattices and integrated superBrownian excursion. Probab. Theory Related Fields, 128, 161–212 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cori R., Vauquelin B.: Planar maps are well labeled trees. Canad. J. Math., 33(5), 1023–1042 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  9. Di Francesco, P., Ginsparg, P. & Zinn-Justin, J., 2D gravity and random matrices. Phys. Rep, 254 (1995).

  10. Duplantier B., Sheffield S.: Liouville quantum gravity and KPZ. Invent. Math., 185, 333–393 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Duquesne T., Le Gall J.F.: Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields, 131, 553–603 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Evans S.N., Pitman J., Winter A.: Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields, 134, 81–126 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fitzsimmons, P., Pitman, J. & Yor, M., Markovian bridges: construction, Palm interpretation, and splicing, in Seminar on Stochastic Processes (Seattle, WA, 1992), Progr. Probab., 33, pp. 101–134. Birkhäuser, Boston, MA, 1993.

  14. Le Gall, J.-F., Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, 1999.

  15. Le Gall J.-F.: The topological structure of scaling limits of large planar maps. Invent. Math., 169, 621–670 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Le Gall J.-F.: Geodesics in large planar maps and in the Brownian map. Acta Math., 205, 287–360 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Le Gall, J.-F., Uniqueness and universality of the Brownian map. To appear in Ann. Probab.

  18. Le Gall J.-F., Miermont G.: Scaling limits of random planar maps with large faces. Ann. Probab., 39(1), 1–69 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Le Gall, J.-F. & Miermont, G., Scaling limits of random trees and planar maps, in Probability and Statistical Physics in Two and More Dimensions (Búzios, 2010), Clay Mathematics Proceedings, 15, pp. 155–211. Amer. Math. Soc., Providence, RI, 2012.

  20. Le Gall J.-F., Paulin F.: Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere. Geom. Funct. Anal., 18, 893–918 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Le Gall J.-F., Weill M.: Conditioned Brownian trees. Ann. Inst. H. Poincaré Probab. Stat., 42, 455–489 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Marckert J.-F., Miermont G.: Invariance principles for random bipartite planar maps. Ann. Probab., 35(5), 1642–1705 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Marckert J.-F., Mokkadem A.: Limit of normalized quadrangulations: the Brownian map. Ann. Probab., 34(6), 2144–2202 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Miermont, G., An invariance principle for random planar maps, in Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities (Nancy, 2006), Discrete Math. Theor. Comput. Sci. Proc., AG, pp. 39–57. Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2006.

  25. Miermont G.: On the sphericity of scaling limits of random planar quadrangulations. Electron. Commun. Probab., 13, 248–257 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Miermont G.: Tessellations of random maps of arbitrary genus. Ann. Sci. Éc. Norm. Supér., 42, 725–781 (2009)

    MathSciNet  MATH  Google Scholar 

  27. Okounkov A.: Random matrices and random permutations. Int. Math. Res. Not., 20, 1043–1095 (2000)

    Article  MathSciNet  Google Scholar 

  28. Petrov, V.V., Sums of Independent Random Variables. Ergebnisse der Mathematik und ihrer Grenzgebiete, 82. Springer, New York, 1975.

  29. Pitman, J., Combinatorial Stochastic Processes. Lecture Notes in Mathematics, 1875. Springer, Berlin–Heidelberg, 2006.

  30. Pitman, J. & Yor, M., Decomposition at the maximum for excursions and bridges of one-dimensional diffusions, in Itô’s Stochastic Calculus and Probability Theory, pp. 293–310. Springer, Tokyo, 1996.

  31. Resnick, S. I., Extreme Values, Regular Variation, and Point Processes. Applied Probability, 4. Springer, New York, 1987.

  32. Revuz, D. & Yor, M., Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften, 293. Springer, Berlin–Heidelberg, 1999.

  33. Schaeffer, G., Conjugaison d’arbres et cartes combinatoires aléatoires. Ph.D. Thesis, Université Bordeaux I, Bordeaux, 1998.

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Authors and Affiliations

  1. Unité de mathématiques pures et appliquées, École Normale Supérieure de Lyon, 46, allée d’Italie, FR-69364, Lyon Cedex 07, France

    Grégory Miermont

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  1. Grégory Miermont
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Miermont, G. The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math 210, 319–401 (2013). https://doi.org/10.1007/s11511-013-0096-8

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  • DOI: https://doi.org/10.1007/s11511-013-0096-8

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