The two uniform infinite quadrangulations of the plane have the same law

Laurent Ménard

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 1, page 190-208
  • ISSN: 0246-0203

Abstract

top
We prove that the uniform infinite random quadrangulations defined respectively by Chassaing–Durhuus and Krikun have the same distribution.

How to cite

top

Ménard, Laurent. "The two uniform infinite quadrangulations of the plane have the same law." Annales de l'I.H.P. Probabilités et statistiques 46.1 (2010): 190-208. <http://eudml.org/doc/241421>.

@article{Ménard2010,
abstract = {We prove that the uniform infinite random quadrangulations defined respectively by Chassaing–Durhuus and Krikun have the same distribution.},
author = {Ménard, Laurent},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random map; random tree; Schaeffer’s bijection; uniform infinite planar quadrangulation; uniform infinite planar tree; random planar maps; random labeled trees; Schaeffer's bijection},
language = {eng},
number = {1},
pages = {190-208},
publisher = {Gauthier-Villars},
title = {The two uniform infinite quadrangulations of the plane have the same law},
url = {http://eudml.org/doc/241421},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Ménard, Laurent
TI - The two uniform infinite quadrangulations of the plane have the same law
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 1
SP - 190
EP - 208
AB - We prove that the uniform infinite random quadrangulations defined respectively by Chassaing–Durhuus and Krikun have the same distribution.
LA - eng
KW - random map; random tree; Schaeffer’s bijection; uniform infinite planar quadrangulation; uniform infinite planar tree; random planar maps; random labeled trees; Schaeffer's bijection
UR - http://eudml.org/doc/241421
ER -

References

top
  1. [1] J. Ambjørn, B. Durhuus and T. Jonsson. Quantum Geometry. Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, 1997. Zbl1096.82500MR1465433
  2. [2] O. Angel. Growth and percolation on the uniform infinite planar triangulation. Geom. Funct. Anal. 13 (2003) 935–974. Zbl1039.60085MR2024412
  3. [3] O. Angel. Scaling of percolation on infinite planar maps, i. Preprint, 2005. Available at http://arxiv.org/abs/math/0501006. 
  4. [4] O. Angel and O. Schramm. Uniform infinite planar triangulations. Comm. Math. Phys. 241 (2003) 191–213. Zbl1098.60010MR2013797
  5. [5] K. B. Athreya and P. E. Ney. Branching Processes. Springer, New York, 1972. Zbl0259.60002MR373040
  6. [6] J. Bouttier, P. Di Francesco and E. Guitter. Planar maps as labeled mobiles. Electron. J. Combin. 11 (2004) 27 pp. (electronic). Zbl1060.05045MR2097335
  7. [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. Zbl1102.60007MR2243873
  8. [8] P. Chassaing and G. Schaeffer. Random planar lattices and integrated superBrownian excursion. Probab. Theory Related Fields 128 (2004) 161–212. Zbl1041.60008MR2031225
  9. [9] R. Cori and B. Vauquelin. Planar maps are well labeled trees. Canad. J. Math. 33 (1981) 1023–1042. Zbl0415.05020MR638363
  10. [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. Zbl1074.60027MR2050691
  11. [11] M. Krikun. Local structure of random quadrangulations. Preprint, 2006. Available at http://arxiv.org/abs/math/0512304. 
  12. [12] J. Lamperti. A new class of probability limit theorems. J. Math. Mech. 11 (1962) 749–772. Zbl0107.35602MR148120
  13. [13] J.-F. Le Gall. The topological structure of scaling limits of large planar maps. Invent. Math. 169 (2007) 621–670. Zbl1132.60013MR2336042
  14. [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. Zbl1166.60006MR2438999
  15. [15] J.-F. Marckert and G. Miermont. Invariance principles for random bipartite planar maps. Ann. Probab. 35 (2007) 1642–1705. Zbl1208.05135MR2349571
  16. [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. Zbl1044.60083MR2033196
  17. [17] J. Neveu. Arbres et processus de Galton–Watson. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986) 199–207. Zbl0601.60082MR850756
  18. [18] G. Schaeffer. Conjugaisons d’arbres et cartes combinatoires aléatoires. Ph.D. thesis, Université de Bordeaux I, 1998. 
  19. [19] W. T. Tutte. A census of planar maps. Canad. J. Math. 15 (1963) 249–271. Zbl0115.17305MR146823

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.