Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Geodesics in Brownian surfaces (Brownian maps)

Jérémie Bettinelli

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We define a class a metric spaces we call Brownian surfaces, arising as the scaling limits of random maps on general orientable surfaces with a boundary and we study the geodesics from a uniformly chosen random point. These metric spaces generalize the well-known Brownian map and our results generalize the properties shown by Le Gall on geodesics in the latter space. We use a different approach based on two ingredients: we first study typical geodesics and then all geodesics by an “entrapment” strategy. In particular, we give geometrical characterizations of some subsets of interest, in terms of geodesics, boundary points and concatenations of geodesics forming a loop that is not homotopic to $0$.


On définit une classe d’espaces métriques aléatoires que nous appelons surfaces browniennes : ces objets apparaissent comme limites d’échelle de cartes aléatoires sur des surfaces orientables à bord générales. Dans un second temps, on étudie les géodésiques émanant d’un point choisi uniformément au hasard. Les surfaces browniennes généralisent la fameuse carte brownienne et nos résultats généralisent les propriétés obtenues par Le Gall sur les géodésiques dans cet espace. On utilise une approche différente reposant sur deux ingrédients : on étudie d’abord les géodésiques aux points typiques et on attrape ensuite les autres points en les « encerclant » par de telles géodésiques. En particulier, on obtient des caractérisations géométriques de certains sous-ensembles d’intérêt en termes de géodésiques, points du bord et concaténations des géodésiques formant une boucle non homotope à $0$.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 2 (2016), 612-646.

Received: 8 October 2014
Accepted: 19 December 2014
First available in Project Euclid: 4 May 2016

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Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60C05: Combinatorial probability 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 57N05: Topology of $E^2$ , 2-manifolds 05C80: Random graphs [See also 60B20] 05C12: Distance in graphs

Brownian surfaces Brownian map Geodesics Random maps Scaling limits Gromov–Hausdorff topology Random metric spaces Bijections


Bettinelli, Jérémie. Geodesics in Brownian surfaces (Brownian maps). Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 2, 612--646. doi:10.1214/14-AIHP666.

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  • [1] C. Abraham. Rescaled bipartite planar maps converge to the Brownian map. Preprint, 2013. Available at arXiv:1312.5959.
  • [2] L. Addario-Berry and M. Albenque. The scaling limit of random simple triangulations and random simple quadrangulations. Preprint, 2013. Available at arXiv:1306.5227.
  • [3] J. Ambjørn and T. G. Budd. Trees and spatial topology change in causal dynamical triangulations. J. Phys. A 46 (31) (2013) 315201, 33.
  • [4] E. G. Begle. Regular convergence. Duke Math. J. 11 (1944) 441–450.
  • [5] J. Bertoin, L. Chaumont and J. Pitman. Path transformations of first passage bridges. Electron. Commun. Probab. 8 (2003) 155–166 (electronic).
  • [6] J. Bettinelli. Scaling limits for random quadrangulations of positive genus. Electron. J. Probab. 15 (2010) 1594–1644.
  • [7] J. Bettinelli. The topology of scaling limits of positive genus random quadrangulations. Ann. Probab. 40 (5) (2012) 1897–1944.
  • [8] J. Bettinelli. Scaling limit of random planar quadrangulations with a boundary. Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015) 432–477.
  • [9] J. Bettinelli, E. Jacob and G. Miermont. The scaling limit of uniform random plane maps, via the Ambjørn–Budd bijection. Electron. J. Probab. 19 (2014) 1–16.
  • [10] J. Bettinelli and G. Miermont. Compact Brownian surfaces I. Brownian disks. Preprint. Available at arXiv:1507.08776.
  • [11] J. Bettinelli and G. Miermont. Compact Brownian surfaces II. The general case. In preparation, 2016.
  • [12] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968.
  • [13] J. Bouttier, P. Di Francesco and E. Guitter. Planar maps as labeled mobiles. Electron. J. Combin. 11 (1) (2004), Research Paper 69, 27 pp. (electronic).
  • [14] D. Burago, Y. Burago and S. Ivanov. A Course in Metric Geometry. Graduate Studies in Mathematics 33. American Mathematical Society, Providence, RI, 2001.
  • [15] G. Chapuy. The structure of unicellular maps, and a connection between maps of positive genus and planar labeled trees. Probab. Theory Related Fields 147 (3–4) (2010) 415–447.
  • [16] G. Chapuy, M. Marcus and G. Schaeffer. A bijection for rooted maps on orientable surfaces. SIAM J. Discrete Math. 23 (3) (2009) 1587–1611.
  • [17] P. Chassaing and G. Schaeffer. Random planar lattices and integrated superBrownian excursion. Probab. Theory Related Fields 128 (2) (2004) 161–212.
  • [18] R. Cori and B. Vauquelin. Planar maps are well labeled trees. Canad. J. Math. 33 (5) (1981) 1023–1042.
  • [19] N. Curien, J.-F. Le Gall and G. Miermont. The Brownian cactus I. Scaling limits of discrete cactuses. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2) (2013) 340–373.
  • [20] T. Duquesne and J.-F. Le Gall. Random trees, Lévy processes and spatial branching processes. Astérisque 281, vi+147 (2002).
  • [21] M. Gromov. Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics 152. Birkhäuser, Boston, MA, 1999. 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.
  • [22] J.-F. Le Gall. Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1999.
  • [23] J.-F. Le Gall. The topological structure of scaling limits of large planar maps. Invent. Math. 169 (3) (2007) 621–670.
  • [24] J.-F. Le Gall. Geodesics in large planar maps and in the Brownian map. Acta Math. 205 (2) (2010) 287–360.
  • [25] J.-F. Le Gall. Uniqueness and universality of the Brownian map. Ann. Probab. 41 (4) (2013) 2880–2960.
  • [26] J.-F. Le Gall and G. Miermont. Scaling limits of random planar maps with large faces. Ann. Probab. 39 (1) (2011) 1–69.
  • [27] J.-F. Le Gall and F. Paulin. Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere. Geom. Funct. Anal. 18 (3) (2008) 893–918.
  • [28] J.-F. Marckert and A. Mokkadem. Limit of normalized quadrangulations: The Brownian map. Ann. Probab. 34 (6) (2006) 2144–2202.
  • [29] G. Miermont. On the sphericity of scaling limits of random planar quadrangulations. Electron. Commun. Probab. 13 (2008) 248–257.
  • [30] G. Miermont. Tessellations of random maps of arbitrary genus. Ann. Sci. Éc. Norm. Supér. (4) 42 (5) (2009) 725–781.
  • [31] G. Miermont. The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210 (2) (2013) 319–401.
  • [32] S. B. Myers. Connections between differential geometry and topology II. Closed surfaces. Duke Math. J. 2 (1) (1936) 95–102.
  • [33] D. Poulalhon and G. Schaeffer. Optimal coding and sampling of triangulations. Algorithmica 46 (3–4) (2006) 505–527.
  • [34] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin, 1999.
  • [35] G. Schaeffer. Conjugaison d’arbres et cartes combinatoires aléatoires. Ph.D. thesis, Université de Bordeaux 1, 1998.
  • [36] G. T. Whyburn. On sequences and limiting sets. Fund. Math. 25 (1935) 408–426.