The Annals of Probability

The topology of scaling limits of positive genus random quadrangulations

Jérémie Bettinelli

Full-text: Open access


We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given $g$, we consider, for every $n\ge1$, a random quadrangulation $\mathfrak{q}_{n}$ 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 metric. As $n$ tends to infinity, this metric space, with distances rescaled by the factor $n^{-1/4}$, 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 limiting space is almost surely homeomorphic to the genus $g$-torus.

Article information

Ann. Probab. Volume 40, Number 5 (2012), 1897-1944.

First available in Project Euclid: 8 October 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 57N05: Topology of $E^2$ , 2-manifolds

Random map random tree regular convergence Gromov topology


Bettinelli, Jérémie. The topology of scaling limits of positive genus random quadrangulations. Ann. Probab. 40 (2012), no. 5, 1897--1944. doi:10.1214/11-AOP675.

Export citation


  • [1] Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19 1–28.
  • [2] Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
  • [3] Begle, E. G. (1944). Regular convergence. Duke Math. J. 11 441–450.
  • [4] Bertoin, J., Chaumont, L. and Pitman, J. (2003). Path transformations of first passage bridges. Electron. Commun. Probab. 8 155–166 (electronic).
  • [5] Bettinelli, J. (2010). Scaling limits for random quadrangulations of positive genus. Electron. J. Probab. 15 1594–1644.
  • [6] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [7] Bousquet-Mélou, M. and Janson, S. (2006). The density of the ISE and local limit laws for embedded trees. Ann. Appl. Probab. 16 1597–1632.
  • [8] Burago, D., Burago, Y. and Ivanov, S. (2001). A Course in Metric Geometry. Graduate Studies in Mathematics 33. Amer. Math. Soc., Providence, RI.
  • [9] Chapuy, G. (2010). The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees. Probab. Theory Related Fields 147 415–447.
  • [10] Chapuy, G., Marcus, M. and Schaeffer, G. (2009). A bijection for rooted maps on orientable surfaces. SIAM J. Discrete Math. 23 1587–1611.
  • [11] Chassaing, P. and Schaeffer, G. (2004). Random planar lattices and integrated superBrownian excursion. Probab. Theory Related Fields 128 161–212.
  • [12] Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque (281) vi+147.
  • [13] Epstein, D. B. A. (1966). Curves on $2$-manifolds and isotopies. Acta Math. 115 83–107.
  • [14] Fitzsimmons, P., Pitman, J. and Yor, M. (1993). Markovian bridges: Construction, Palm interpretation, and splicing. In Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992). Progress in Probability 33 101–134. Birkhäuser, Boston, MA.
  • [15] Gromov, M. (2007). Metric Structures for Riemannian and Non-Riemannian Spaces, English ed. Birkhäuser, Boston, MA.
  • [16] Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhäuser, Basel.
  • [17] Le Gall, J.-F. (2005). Random trees and applications. Probab. Surv. 2 245–311 (electronic).
  • [18] Le Gall, J.-F. (2007). The topological structure of scaling limits of large planar maps. Invent. Math. 169 621–670.
  • [19] Le Gall, J.-F. (2010). Geodesics in large planar maps and in the Brownian map. Acta Math. 205 287–360.
  • [20] Le Gall, J.-F. and Paulin, F. (2008). Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere. Geom. Funct. Anal. 18 893–918.
  • [21] Marckert, J.-F. and Mokkadem, A. (2006). Limit of normalized quadrangulations: The Brownian map. Ann. Probab. 34 2144–2202.
  • [22] Miermont, G. (2008). On the sphericity of scaling limits of random planar quadrangulations. Electron. Commun. Probab. 13 248–257.
  • [23] Miermont, G. (2009). Tessellations of random maps of arbitrary genus. Ann. Sci. Éc. Norm. Supér. (4) 42 725–781.
  • [24] Neveu, J. (1986). Arbres et processus de Galton–Watson. Ann. Inst. H. Poincaré Probab. Stat. 22 199–207.
  • [25] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [26] Whyburn, G. T. (1935). Regular convergence and monotone transformations. Amer. J. Math. 57 902–906.