The two uniform infinite quadrangulations of the plane have the same law
Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 1, p. 190-208
We prove that the uniform infinite random quadrangulations defined respectively by Chassaing-Durhuus and Krikun have the same distribution.
On démontre que les quadrangulations aléatoires infinies uniformes définies respectivement par Chassaing-Durhuus et par Krikun ont la même loi.
DOI : https://doi.org/10.1214/09-AIHP313
Classification:  60C05,  60J80,  05C30
Keywords: random map, random tree, Schaeffer's bijection, uniform infinite planar quadrangulation, uniform infinite planar tree
@article{AIHPB_2010__46_1_190_0,
     author = {M\'enard, Laurent},
     title = {The two uniform infinite quadrangulations of the plane have the same law},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {1},
     year = {2010},
     pages = {190-208},
     doi = {10.1214/09-AIHP313},
     zbl = {1201.60009},
     mrnumber = {2641776},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2010__46_1_190_0}
}
Ménard, Laurent. The two uniform infinite quadrangulations of the plane have the same law. Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 1, pp. 190-208. doi : 10.1214/09-AIHP313. http://www.numdam.org/item/AIHPB_2010__46_1_190_0/

[1] J. Ambjørn, B. Durhuus and T. Jonsson. Quantum Geometry. Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, 1997. | MR 1465433 | Zbl 1096.82500

[2] O. Angel. Growth and percolation on the uniform infinite planar triangulation. Geom. Funct. Anal. 13 (2003) 935-974. | MR 2024412 | Zbl 1039.60085

[3] O. Angel. Scaling of percolation on infinite planar maps, i. Preprint, 2005. Available at http://arxiv.org/abs/math/0501006.

[4] O. Angel and O. Schramm. Uniform infinite planar triangulations. Comm. Math. Phys. 241 (2003) 191-213. | MR 2013797 | Zbl 1098.60010

[5] K. B. Athreya and P. E. Ney. Branching Processes. Springer, New York, 1972. | MR 373040 | Zbl 0259.60002

[6] J. Bouttier, P. Di Francesco and E. Guitter. Planar maps as labeled mobiles. Electron. J. Combin. 11 (2004) 27 pp. (electronic). | MR 2097335 | Zbl 1060.05045

[7] P. Chassaing and B. Durhuus. Local limit of labeled trees and expected volume growth in a random quadrangulation. Ann. Probab. 34 (2006) 879-917. | MR 2243873 | Zbl 1102.60007

[8] P. Chassaing and G. Schaeffer. Random planar lattices and integrated superBrownian excursion. Probab. Theory Related Fields 128 (2004) 161-212. | MR 2031225 | Zbl 1041.60008

[9] R. Cori and B. Vauquelin. Planar maps are well labeled trees. Canad. J. Math. 33 (1981) 1023-1042. | MR 638363 | Zbl 0415.05020

[10] M. Krikun. A uniformly distributed infinite planar triangulation and a related branching process. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 307 (2004) 141-174, 282-283. | MR 2050691 | Zbl 1074.60027

[11] M. Krikun. Local structure of random quadrangulations. Preprint, 2006. Available at http://arxiv.org/abs/math/0512304.

[12] J. Lamperti. A new class of probability limit theorems. J. Math. Mech. 11 (1962) 749-772. | MR 148120 | Zbl 0107.35602

[13] J.-F. Le Gall. The topological structure of scaling limits of large planar maps. Invent. Math. 169 (2007) 621-670. | MR 2336042 | Zbl 1132.60013

[14] J.-F. Le Gall and F. Paulin. Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere. Geom. Funct. Anal. 18 (2008) 893-918. | MR 2438999 | Zbl 1166.60006

[15] J.-F. Marckert and G. Miermont. Invariance principles for random bipartite planar maps. Ann. Probab. 35 (2007) 1642-1705. | MR 2349571 | Zbl 1208.05135

[16] J.-F. Marckert and A. Mokkadem. States spaces of the snake and its tour-convergence of the discrete snake. J. Theoret. Probab. 16 (2004) 1015-1046. | MR 2033196 | Zbl 1044.60083

[17] J. Neveu. Arbres et processus de Galton-Watson. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986) 199-207. | Numdam | MR 850756 | Zbl 0601.60082

[18] G. Schaeffer. Conjugaisons d'arbres et cartes combinatoires aléatoires. Ph.D. thesis, Université de Bordeaux I, 1998.

[19] W. T. Tutte. A census of planar maps. Canad. J. Math. 15 (1963) 249-271. | MR 146823 | Zbl 0115.17305