Invariance principles for random bipartite planar maps

Annals of Probability

Volumn 35, Issue 5, 2007, Pages 1642-1705

  Marckert, Jean François ^a   Miermont, Grégory ^a  

^a CNRS   (France)

Author keywords

Brownian map; Brownian snake; Invariance principle; Labeled mobiles; Random planar maps; Spatial Galton Watson trees

Indexed keywords

EID: 38749135683     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/009117906000000908     Document Type: Article

References (24)

1. ALDOUS, D. J. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (Durham, 1990). London Math. Soc. Lecture Note Ser. 167 23-70. Cambridge Univ. Press. MR1166406
2. ALDOUS, D. J. (1993). The continuum random, tree. III. Ann. Probab. 21 248-289. MR1207226
3. AMBJØRN, J., DURHUUS, B. and JONSSON, T. (1997). Quantum Geometry. A Statistical Field Theory Approach. Cambridge Univ. Press. MR.1465433
4. ANGEL, O. (2003). Growth and percolation on the uniform infinite planar triangulation. Geom. Funct. Anal. 13 935-974. MR2024412
5. ANGEL, O. and SCHRAMM., O. (2003). Uniform infinite planar triangulations. Comm. Math. Phys. 241 191-213. MR2013797
6. BOUTTIER, J., DI FRANCESCO, P. and QUITTER, E. (2004). Planar maps as labeled mobiles. Electron. J. Combin. 11 1-27. MR2097335
7. CHASSAING, P. and DURHUUS, B. (2006). Local limit of labelled trees and expected, volume growth in a random quadrangulation. Ann. Probab. 34 879-917. MR2243873
8. CHASSAING, P. and SCHAEFFER, G. (2004). Random planar lattices and integrated super-Brownian excursion. Probab. Theory Related Fields 128 161-212. MR2031225
9. DUQUESNE, T. (2003). A limit theorem for the contour process of conditioned Galton-Watson trees. Ann. Probab. 31 996-1027. MR1964956
10. DUQUESNE, T. and. LE GALL, J.-F. (2002). Random, trees, Levy processes and spatial branching processes. Astérisque 281. MR 1954248
11. FELLER, W. (1971). An Introduction to Probability Theory and Its Applications II, 2nd ed. Wiley, New York. MR0270403
12. GITTENBERGER, B. (2004). 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
13. JANSON, S. and MARCKERT, J.-F. (2005). Convergence of discrete snakes. J. Theoret. Probab. 18 615-645. MR2167644
14. KURTZ, T., LYONS, R., PEMANTLE, R. and PERES, Y. (.1997). A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes. In Classical and Modern Branching Processes (Minneapolis, MN, 1994). IMA Vol. Math. Appl. 84 181-185. Springer, New York. MR 1601737
15. LE GALL, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhäuser, Basel. MR1714707
16. LE GALL, J.-F. (2006). Conditional limit theorem for tree-indexed random walks. Stochastic Process. Appl. 116 539-567. MR2205115
17. LE GALL, J.-F. and WEILL, M. (2006). Conditioned Brownian trees. Ann. Inst. H. Poincaré Probab. Statist. 42 455-489. MR2242956
18. 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. MR 1989446
19. MARCKERT, J.-F. and MOKKADEM, A. (2004). States spaces of the snake and its tour-convergence of the discrete snake. J. Theoret. Probab. 16 1015-1046. MR2033196
20. MARCKERT, J.-F. and MOKKADEM, A. (2006). Limit of normalized random quadrangulations: The Brownian map. Ann. Probab. 34 2144-2202.
21. PETROV, V. V. (1995). Limit Theorems of Probability Theory. Oxford. Univ. Press, New York. MR 1353441
22. PITMAN, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Springer, Berlin. MR2245368
23. SCHAEFFER, G. (1998). Conjugaison d'arbres et cartes combinatoires aléatoires. Ph.D. thesis, Univ. Bordeaux I. Available at http://www.lix.polytechmque.fr/~schaeffe/Bi.blio/PhD-Schaeffer.ps.
24. STROOCK, D. W. (1993). Probability Theory, an Analytic View. Cambridge Univ. Press. MR1267569