Annals of Probability

Local limit of labeled trees and expected volume growth in a random quadrangulation

Philippe Chassaing and Bergfinnur Durhuus

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Exploiting a bijective correspondence between planar quadrangulations and well-labeled trees, we define an ensemble of infinite surfaces as a limit of uniformly distributed ensembles of quadrangulations of fixed finite volume. The limit random surface can be described in terms of a birth and death process and a sequence of multitype Galton–Watson trees. As a consequence, we find that the expected volume of the ball of radius r around a marked point in the limit random surface is Θ(r4).

Article information

Ann. Probab., Volume 34, Number 3 (2006), 879-917.

First available in Project Euclid: 27 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 05C30: Enumeration in graph theory 05C05: Trees 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Random surface quadrangulation expected volume growth well-labeled trees Galton–Watson trees birth and death process quantum gravity


Chassaing, Philippe; Durhuus, Bergfinnur. Local limit of labeled trees and expected volume growth in a random quadrangulation. Ann. Probab. 34 (2006), no. 3, 879--917. doi:10.1214/009117905000000774.

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  • Aldous, D. (1991). Asymptotic fringe distributions for general families of random trees. Ann. Appl. Probab. 1 228–266.
  • Aldous, D. (1993). Tree-based models for random distribution of mass. J. Statist. Phys. 73 625–641.
  • Ambjørn, J., Durhuus, B. and Fröhlich, J. (1985). Diseases of triangulated random surfaces. Nuclear Phys. B 257 433–449.
  • Ambjørn, J., Durhuus, B. and Jónsson, T. (1991). Three-dimensional simplicial quantum gravity and generalized matrix models. Mod. Phys. Lett. A 6 1133–1147.
  • Ambjørn, J., Durhuus, B. and Jónsson, 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. (2003). Growth and percolation on the uniform infinite planar triangulation. Geom. Funct. Anal. 13 935–974.
  • Angel, O. and Schramm, O. (2003). Uniform infinite planar triangulations. Comm. Math. Phys. 241 191–214.
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Bouttier, J., Di Francesco, P. and Guitter, E. (2003). Geodesic distance in planar graphs. Nuclear Phys. B 663 535–567.
  • Brezin, E., Itzykson, C., Parisi, G. and Zuber, J.-B. (1978). Planar diagrams. Comm. Math. Phys. 59 1035–1047.
  • 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. (1985). Planar diagrams, two-dimensinal lattice gravity and surface models. Nuclear Phys. B 257 45–58.
  • Durhuus, B. (2003). Probabilistic aspects of planar trees and surfaces. Acta Phys. Polonica B 34 4795–4811.
  • Gillet, F. (2003). Études d'algorithmes stochastiques et arbres. Ph.D. thesis, Univ. Henri Poincaré, Nancy. Available at
  • Flajolet, P. and Odlyzko, A. M. (1990). Singularity analysis of generating functions. SIAM J. Discrete Math. 3 216–240.
  • Flajolet, P. and Sedgewick, R. (2006). Analytic combinatorics. Available at
  • Janson, S. (2002). Ideals in a forest, one-way infinite binary trees and the contraction method. In Mathematics and Computer Science (B. Chauvin, P. Flajolet, D. Gardy and A. Mokkadem, eds.) 2 393–414. Birkhäuser, Basel.
  • Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, 2nd ed. Academic Press, New York.
  • Kazakov, V. A., Kostov, I. and Migdal, A. A. (1985). Critical properties of randomly triangulated planar random surfaces. Phys. Lett. B 157 295–300.
  • Kurtz, T., Lyons, R., Pemantle, R. and Peres, Y. (1997). A conceptual proof of the Kesten–Stigum theorem for multi-type branching processes. In Classical and Modern Branching Processes (K. B. Athreya and P. Jagers, eds.) 181–185. Springer, New York.
  • Le Gall, J.-F. (1986). An elementary approach to Williams' decomposition theorems. Séminaire de Probabilités XX. Lecture Notes in Math. 1204 447–464. Springer, Berlin.
  • Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of L log L criteria for mean behavior of branching processes. Ann. Probab. 23 1125–1138.
  • Newman, M. H. A. (1992). Elements of the Topology of Plane Sets of Points. (Reprint of the second edition) Dover, New York.
  • Rudin, W. (1987). Real and Complex Analysis, 3rd ed. McGraw–Hill, New York.
  • Schaeffer, G. (1998). Conjugaison d'arbres et cartes combinatoires aléatoires. Ph.D. thesis, Université Bordeaux I, 1998, Bordeaux.
  • Slade, G. (2002). Scaling limits and super-Brownian motion. Notices Amer. Math. Soc. 49 1056–1067.
  • Tutte, W. T. (1963). A census of planar maps. Canad. J. Math. 15 249–271.