The Annals of Probability

Limit of normalized quadrangulations: The Brownian map

Jean-François Marckert and Abdelkader Mokkadem

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Consider qn a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with n faces. In this paper we show that, when n goes to +∞, qn suitably normalized converges weakly in a certain sense to a random limit object, which is continuous and compact, and that we name the Brownian map. The same result is shown for a model of rooted quadrangulations and for some models of rooted quadrangulations with random edge lengths. A metric space of rooted (resp. pointed) abstract maps that contains the model of discrete rooted (resp. pointed) quadrangulations and the model of the Brownian map is defined. The weak convergences hold in these metric spaces.

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Ann. Probab., Volume 34, Number 6 (2006), 2144-2202.

First available in Project Euclid: 13 February 2007

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Zentralblatt MATH identifier

Primary: 60F99: None of the above, but in this section 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems

Brownian map quadrangulation planar map limit theorem weak convergence trees abstract maps


Marckert, Jean-François; Mokkadem, Abdelkader. Limit of normalized quadrangulations: The Brownian map. Ann. Probab. 34 (2006), no. 6, 2144--2202. doi:10.1214/009117906000000557.

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  • Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (M. T. Barlow and N. H. Bingham, eds.) 23--70. Cambridge Univ. Press.
  • Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248--289.
  • Aldous, D. and Pitman, P. (1999). A family of random trees with random edge lengths. Random Structures Algorithms 15 176--195.
  • Aldous, D. and Pitman, J. (2000). Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent. Probab. Theory Related Fields 118 455--482.
  • Aldous, D., Miermont, G. and Pitman, J. (2004). The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin's local time identity. Probab. Theory Related Fields 129 182--218.
  • Ambjørn, J., Durhuus, B. and Jonsson, T. (1997). Quantum Geometry. A Statistical Field Theory Approach. Cambridge Univ. Press.
  • Ambjørn, J. and Watabiki, Y. (1995). Scaling in quantum gravity. Nuclear Phys. B 445 129--142.
  • Angel, O. and Schramm, O. (2003). Uniform infinite planar triangulations. Comm. Math. Phys. 241 191--213.
  • Bouttier, J., Di Francesco, P. and Guitter, E. (2003). Statistics of planar graphs viewed from a vertex: A study via labeled trees. Nuclear Phys. B 675 631--660.
  • Bender, E. A., Compton, K. J. and Richmond, L. B. (1999). 0--1 laws for maps. Random Structures Algorithms 14 215--237.
  • Bender, E. A., Richmond, L. B. and Wormald, N. C. (1995). Largest 4-connected components of 3-connected planar triangulations. Random Structure Algorithms 7 273--285.
  • Brézin, E., Itzykson, C., Parisi, G. and Zuber, J. B. (1978). Planar diagrams. Comm. Math. Phys. 59 35--51.
  • Camarri, M. and Pitman, J. (2000). Limit distributions and random trees derived from the birthday problem with unequal probabilities. Electron. J. Probab. 5 1--18.
  • Chassaing, P. and Durhuus, B. (2006). Local limit of labeled trees and expected volume growth in a random quadrangulation. Ann. Probab. 34 879--917.
  • Chassaing, P. and Schaeffer, G. (2004). Random planar lattices and integrated superBrownian excursion. Probab. Theory Related Fields 128 161--212.
  • Cori, R. and Vauquelin, B. (1981). Planar maps are well-labeled trees. Canad. J. Math. 33 1023--1042.
  • David, F. (1995). Simplicial quantum gravity and random lattices. In Gravitation and Quantizations (Les Houches, 1992) 679--749. North-Holland, Amsterdam.
  • Duquesne, T. and Le Gall, J. F. (2002). Random Trees, Lévy Processes and Spatial Branching Processes. Astérisque 281. Soc. Math. de France, Paris.
  • Evans, S. N., Pitman, J. and Winter, A. (2006). Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 81--126.
  • Gao, Z. and Richmond, L. B. (1994). Root vertex valency distributions of rooted maps and rooted triangulations. European J. Combin. 15 483--490.
  • Gittenberger, B. (2003). A note on ``State spaces of the snake and its tour---Convergence of the discrete snake'' by J.-F. Marckert and A. Mokkadem. J. Theoret. Probab. 16 1063--1067.
  • Janson, S. and Marckert, J.-F. (2003). Convergence of discrete snake. J. Theoret. Probab. 18 615--647.
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • Krikun, M. (2003). Uniform infinite planar triangulation and related time-reversed critical branching process. J. Math. Sci. 131 5520--5537.
  • Le Gall, J.-F. and Le Jan, Y. (1998). Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 213--252.
  • Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhäuser, Basel.
  • Le Gall, J.-F. and Weill, M. (2006). Conditioned Brownian trees. Ann. Inst. H. Poincaré Probab. Statist. 42 455--489.
  • Marckert, J.-F. and Mokkadem, A. (2003). The depth first processes of Galton--Watson trees converge to the same Brownian excursion. Ann. Probab. 31 1655--1678.
  • Marckert, J.-F. and Mokkadem, A. (2003). States spaces of the snake and of its tour---Convergence of the discrete snake. J. Theoret. Probab. 16 1015--1046.
  • Petrov, V. V. (1995). Limit Theorems of Probability Theory. Sequences of Independent Random Variables. Oxford Univ. Press.
  • Pitman, J. (2006). Combinatorial Stochastic Processes. Springer, Berlin. MR2245368
  • Richmond, L. B. and Wormald, N. C. (1995). Almost all maps are asymmetric. J. Combin. Theory Ser. B 63 1--7.
  • Schaeffer, G. (1998). Conjugaison d'arbres et cartes combinatoires aléatoires. Ph.D. thesis, Univ. Bordeaux I. Available at
  • Tutte, W. T. (1963). A census of planar maps. Canad. J. Math. 15 249--271.
  • Tutte, W. T. (1980). On the enumeration of convex polyhedra. J. Combin. Theory Ser. B 28 105--126.
  • Walkup, D. W. (1972). The number of plane trees. Mathematika 19 200--204.
  • Watabiki, Y. (1995). Construction of non-critical string field theory by transfer matrix formalism in dynamical triangulation. Nuclear Phys. B 441 119--163.