Limit of normalized quadrangulations: The brownian map

Annals of Probability

Volumn 34, Issue 6, 2006, Pages 2144-2202

  Marckert, Jean François ^a   Mokkadem, Abdelkader ^a  

^a LABORATOIRE DES SCIENCES DU CLIMAT ET DE L ENVIRONNEMENT   (France)

Author keywords

Abstract maps; Brownian map; Limit theorem; Planar map; Quadrangulation; Trees; Weak convergence

Indexed keywords

EID: 33947505882     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/009117906000000557     Document Type: Article

References (37)

1. ALDOUS, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (M. T. Barlow and N. H. Bingham, eds.) 23-70. Cambridge Univ. Press. MR1166406
2. ALDOUS, D. (1993). The continuum random tree. III. Ann. Probab. 21 248-289. MR1207226
3. ALDOUS, D. and PITMAN, P. (1999). A family of random trees with random edge lengths. Random Structures Algorithms 15 176-195. MR1704343
4. ALDOUS, D. and PITMAN, J. (2000). Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent. Probab. Theory Related Fields 118 455-482. MR1808372
5. ALDOUS, D., MIERMONT, G. and PITMAN, J. (2004). The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin's local time identity. Probab. Theory Related Fields 129 182-218. MR2063375
6. AMBJØRN, J., DURHUUS, B. and JONSSON, T. (1997). Quantum Geometry. A Statistical Field Theory Approach. Cambridge Univ. Press. MR1465433
7. AMBJØRN, J. and WATABIKI, Y. (1995). Scaling in quantum gravity. Nuclear Phys. B 445 129-142. MR1338099
8. ANGEL, O. and SCHRAMM, O. (2003). Uniform infinite planar triangulations. Comm. Math. Phys. 241 191-213. MR2013797
9. BOUTTIER, J., Di FRANCESCO, P. and QUITTER, E. (2003). Statistics of planar graphs viewed from a vertex: A study via labeled trees. Nuclear Phys. B 675 631-660. MR2018892
10. BENDER, E. A., COMPTON, K. J. and RICHMOND, L. B. (1999). 0-1 laws for maps. Random Structures Algorithms 14 215-237. MR1680232
11. BENDER, E. A., RICHMOND, L. B. and WORMALD, N. C. (1995). Largest 4-connectedcomponents of 3-connected planar triangulations. Random Structure Algorithms 7 273-285. MR1369067
12. BRÉZIN, E., ITZYKSON, C., PARISI, G. and ZUBER, J. B. (1978). Planar diagrams. Comm. Math. Phys. 59 35-51. MR0471676
13. CAMARRI, M. and PITMAN, J. (2000). Limit distributions and random trees derived from the birthday problem with unequal probabilities. Electron. J. Probab. 5 1-18. MR1741774
14. CHASSAING, P. and DURHUUS, B. (2006). Local limit of labeled trees and expected volume growth in a random quadrangulation. Ann. Probab. 34 879-917. MR2243873
15. CHASSAING, P. and SCHAEFFER, G. (2004). Random planar lattices and integrated super-Brownian excursion. Probab. Theory Related Fields 128 161-212. MR2031225
16. CORI, R. and VAUQUELIN, B. (1981). Planar maps are well-labeled trees. Canad. J. Math. 33 1023-1042. MR0638363
17. DAVID, F. (1995). Simplicial quantum gravity and random lattices. In Gravitation and Quantizations (Les Hauches, 1992) 679-749. North-Holland, Amsterdam. MR1461284
18. DUQUESNE, T. and LE GALL, J. F. (2002). Random Trees, Lévy Processes and Spatial Branching Processes. Astérisque 281. Soc. Math, de France, Paris. MR1954248
19. EVANS, S. N., PITMAN, J. and WINTER, A. (2006). Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 81-126. MR2221786
20. GAO, Z. and RICHMOND, L. B. (1994). Root vertex valency distributions of rooted maps and rooted triangulations. European J. Combin. 15 483-490. MR1292958
21. GITTENBERGER, B. (2003). A note on "State spaces of the snake and its tour-Convergence of the discrete snake" by J.-F. Marckert and A. Mokkadem. J. Theoret. Probab. 16 1063-1067. MR2033198
22. JANSON, S. and MARCKERT, J.-F. (2003). Convergence of discrete snake. J. Theoret. Probab. 18615-647. MR2167644
23. KALLENBERG, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York. MR1876169
24. KRIKUN, M. (2003). Uniform infinite planar triangulation and related time-reversed critical branching process. J. Math. Sci. 131 5520-5537.
25. LE GALL, J.-F. and LE JAN, Y. (1998). Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 213-252. MR 1617047
26. LE GALL, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhäuser, Basel. MR1714707
27. LE GALL, J.-F. and WEILL, M. (2006). Conditioned Brownian trees. Ann. Inst. H. Poincaré Probab. Statist. 42 455-489. MR2242956
28. MARCKERT, J.-F. and MOKKADEM, A. (2003). The depth first processes of Galton-Watson trees converge to the same Brownian excursion. Ann. Probab. 311655-1678. MR1989446
29. MARCKERT, J.-F. and MOKKADEM, A. (2003). States spaces of the snake and of its tour - Convergence of the discrete snake. J. Theoret. Probab. 16 1015-1046. MR2033196
30. PETROV, V. V. (1995). Limit Theorems of Probability Theory. Sequences of Independent Random Variables. Oxford Univ. Press. MR1353441
31. PITMAN, J. (2006). Combinatorial Stochastic Processes. Springer, Berlin. MR2245368
32. RICHMOND, L. B. and WORMALD, N. C. (1995). Almost all maps are asymmetric. J. Combin. Theory Ser. B 63 1-7. MR1309353
33. SCHAEFFER, G. (1998). Conjugaison d'arbres et cartes combinatoires aléatoires. Ph.D. thesis, Univ. Bordeaux I. Available at http://www.lix.polytechnique.fr/~schaeffe/Biblio/PHD-Schaeffer.ps
34. TUTTE, W. T. (1963). A census of planar maps. Canad. J. Math. 15 249-271. MRO146823
35. TUTTE, W. T. (1980). On the enumeration of convex polyhedra. J. Combin. Theory Ser. B 28 105-126. MR0572468
36. WALKUP, D. W. (1972). The number of plane trees. Mathematika 19 200-204. MR0376411
37. WATABIKI, Y. (1995). Construction of non-critical string field theory by transfer matrix formalism in dynamical triangulation. Nuclear Phys. B 441 119-163. MR1329946