## The Annals of Probability

### The scaling limit of random simple triangulations and random simple quadrangulations

#### Abstract

Let $M_{n}$ be a simple triangulation of the sphere $\mathbb{S}^{2}$, drawn uniformly at random from all such triangulations with $n$ vertices. Endow $M_{n}$ with the uniform probability measure on its vertices. After rescaling graph distance by $(3/(4n))^{1/4}$, the resulting random measured metric space converges in distribution, in the Gromov–Hausdorff–Prokhorov sense, to the Brownian map. In proving the preceding fact, we introduce a labelling function for the vertices of $M_{n}$. Under this labelling, distances to a distinguished point are essentially given by vertex labels, with an error given by the winding number of an associated closed loop in the map. We establish similar results for simple quadrangulations.

#### Article information

Source
Ann. Probab., Volume 45, Number 5 (2017), 2767-2825.

Dates
Revised: April 2016
First available in Project Euclid: 23 September 2017

https://projecteuclid.org/euclid.aop/1506132026

Digital Object Identifier
doi:10.1214/16-AOP1124

Mathematical Reviews number (MathSciNet)
MR3706731

Zentralblatt MATH identifier
06812193

#### Citation

Addario-Berry, Louigi; Albenque, Marie. The scaling limit of random simple triangulations and random simple quadrangulations. Ann. Probab. 45 (2017), no. 5, 2767--2825. doi:10.1214/16-AOP1124. https://projecteuclid.org/euclid.aop/1506132026

#### References

• [1] Albenque, M. and Poulalhon, D. (2015). A generic method for bijections between blossoming trees and planar maps. Electron. J. Combin. 22 Paper 2.38, 44.
• [2] Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (Durham, 1990). London Mathematical Society Lecture Note Series 167 23–70. Cambridge Univ. Press, Cambridge.
• [3] Ambjørn, J., Durhuus, B. and Jonsson, T. (1997). Quantum Geometry: A Statistical Field Theory Approach. Cambridge Univ. Press, Cambridge.
• [4] Banderier, C., Flajolet, P., Schaeffer, G. and Soria, M. (2001). Random maps, coalescing saddles, singularity analysis, and Airy phenomena. Random Structures Algorithms 19 194–246.
• [5] Beltran, J. and Le Gall, J.-F. (2013). Quadrangulations with no pendant vertices. Bernoulli 19 1150–1175.
• [6] Bernardi, O. and Fusy, É. (2012). A bijection for triangulations, quadrangulations, pentagulations, etc. J. Combin. Theory Ser. A 119 218–244.
• [7] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
• [8] Bouttier, J. and Guitter, E. (2010). Distance statistics in quadrangulations with no multiple edges and the geometry of minbus. J. Phys. A 43 205207, 31.
• [9] Brown, W. G. (1964). Enumeration of triangulations of the disk. Proc. London Math. Soc. (3) 14 746–768.
• [10] Brown, W. G. (1965). Enumeration of quadrangular dissections of the disk. Canad. J. Math. 17 302–317.
• [11] Burago, D., Burago, Y. and Ivanov, S. (2001). A Course in Metric Geometry. Graduate Studies in Mathematics 33. Amer. Math. Soc., Providence, RI.
• [12] Duplantier, B. and Sheffield, S. (2011). Liouville quantum gravity and KPZ. Invent. Math. 185 333–393.
• [13] Evans, S. N. and Winter, A. (2006). Subtree prune and regraft: A reversible real tree-valued Markov process. Ann. Probab. 34 918–961.
• [14] Fusy, É. (2010). Combinatoire des cartes planaires et applications algorithmiques. Ph.D. thesis, École Polytechnique.
• [15] Fusy, É., Poulalhon, D. and Schaeffer, G. (2005). Dissections and trees, with applications to optimal mesh encoding and to random sampling. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms 690–699 (electronic). ACM, New York.
• [16] Garban, C. (2011–2012). Quantum gravity and the KPZ formula. In Séminaire Bourbaki, Vol. 64.
• [17] Janson, S. and Marckert, J.-F. (2005). Convergence of discrete snakes. J. Theoret. Probab. 18 615–647.
• [18] Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhäuser, Basel.
• [19] Le Gall, J.-F. (2005). Random trees and applications. Probab. Surv. 2 245–311.
• [20] Le Gall, J.-F. (2007). The topological structure of scaling limits of large planar maps. Invent. Math. 169 621–670.
• [21] Le Gall, J.-F. (2010). Geodesics in large planar maps and in the Brownian map. Acta Math. 205 287–360.
• [22] Le Gall, J.-F. (2013). Uniqueness and universality of the Brownian map. Ann. Probab. 41 2880–2960.
• [23] 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.
• [24] Marckert, J.-F. (2008). The lineage process in Galton–Watson trees and globally centered discrete snakes. Ann. Appl. Probab. 18 209–244.
• [25] Marckert, J.-F. and Miermont, G. (2007). Invariance principles for random bipartite planar maps. Ann. Probab. 35 1642–1705.
• [26] Marckert, J.-F. and Mokkadem, A. (2006). Limit of normalized quadrangulations: The Brownian map. Ann. Probab. 34 2144–2202.
• [27] Miermont, G. (2008). On the sphericity of scaling limits of random planar quadrangulations. Electron. Commun. Probab. 13 248–257.
• [28] Miermont, G. (2009). Tessellations of random maps of arbitrary genus. Ann. Sci. Éc. Norm. Supér. (4) 42 725–781.
• [29] Miermont, G. (2013). The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210 319–401.
• [30] Ossona de Mendez, P. (1994). Orientations bipolaires. Ph.D. thesis, École des Hautes Etudes en Sciences Sociales, Paris.
• [31] Poulalhon, D. and Schaeffer, G. (2006). Optimal coding and sampling of triangulations. Algorithmica 46 505–527.
• [32] Schnyder, W. (1989). Planar graphs and poset dimension. Order 5 323–343.
• [33] Stephenson, K. (2005). Introduction to Circle Packing: The Theory of Discrete Analytic Functions. Cambridge Univ. Press, Cambridge.