Convergence to the brownian Web for a generalization of the drainage network model
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 3, p. 899-919

We introduce a system of one-dimensional coalescing nonsimple random walks with long range jumps allowing paths that can cross each other and are dependent even before coalescence. We show that under diffusive scaling this system converges in distribution to the Brownian Web.

Nous introduisons un système de marches aléatoires coalescentes unidimensionnelles, avec des sauts à longue portée. Ce système autorise des chemins qui se croisent, et qui sont dépendants, même avant leur coalescence. Après une renormalisation diffusive, nous montrons que ce système converge en loi vers le réseau brownien.

DOI : https://doi.org/10.1214/13-AIHP544
Classification:  60K35
Keywords: brownian web, coalescing brownian motions, coalescing random walks, drainage network, scaling limit, invariance principle, interacting particle systems
@article{AIHPB_2014__50_3_899_0,
     author = {Coletti, Cristian and Valle, Glauco},
     title = {Convergence to the brownian Web for a generalization of the drainage network model},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {3},
     year = {2014},
     pages = {899-919},
     doi = {10.1214/13-AIHP544},
     zbl = {1296.60080},
     mrnumber = {3224293},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_3_899_0}
}
Coletti, Cristian; Valle, Glauco. Convergence to the brownian Web for a generalization of the drainage network model. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 3, pp. 899-919. doi : 10.1214/13-AIHP544. http://www.numdam.org/item/AIHPB_2014__50_3_899_0/

[1] R. Arratia. Coalescing Brownian motions and the voter model on . Unpublished partial manuscript, 1981.

[2] R. Arratia. Limiting point processes for rescalings of coalescing and annihilating random walks on d . Ann. Probab. 9 (1981) 909-936. | MR 632966 | Zbl 0496.60098

[3] S. Belhaouari, T. Mountford, R. Sun and G. Valle. Convergence results and sharp estimates for the voter model interfaces. Electron. J. Probab. 11 (2006) 768-801. | MR 2242663 | Zbl 1113.60092

[4] C. Coletti, L. R. Fontes and E. Dias. Scaling limit for a drainage network model. J. Appl. Probab. 46 (2009) 1184-1197. | MR 2582714 | Zbl 1186.60104

[5] R. Durrett. Probability: Theory and Examples, 2nd edition. Duxbury Press, Belmont, CA, 1996. | MR 1609153 | Zbl 1202.60002

[6] P. A. Ferrari, L. R. G. Fontes and X. Y. Wu. Two dimensional Poisson trees converge to the Brownian web. Ann. Inst. Henri Poincaré Probab. Stat. 41 (2005) 851-858. | Numdam | MR 2165253 | Zbl 1073.60094

[7] L. R. G. Fontes, M. Isopi, C. M. Newman and K. Ravishankar. The Brownian web. Proc. Natl. Acad. Sci. USA 99 (2002) 15888-15893. | MR 1944976 | Zbl 1069.60068

[8] L. R. G. Fontes, M. Isopi, C. M. Newman and K. Ravishankar. The Brownian web: Characterization and convergence. Ann. Probab. 32 (2004) 2857-2883. | MR 2094432 | Zbl 1105.60075

[9] L. R. G. Fontes, M. Isopi, C. M. Newman and K. Ravishankar. Coarsening, nucleation, and the marked Brownian web. Ann. Inst. Henri Poincaré Probab. Stat. 42 (2006) 37-60. | Numdam | MR 2196970 | Zbl 1087.60072

[10] S. Gangopadhyay, R. Roy and A. Sarkar. Random oriented trees: A model of drainage networks. Ann. App. Probab. 14 (2004) 1242-1266. | MR 2071422 | Zbl 1047.60098

[11] C. M. Newman, K. Ravishankar and R. Sun. Convergence of coalescing nonsimple random walks to the Brownian web. Electron. J. Probab. 10 (2005) 21-60. | MR 2120239 | Zbl 1067.60099

[12] R. Sun. Convergence of coalescing nonsimple random walks to the Brownian web. Ph.D. thesis, Courant Institute of Mathematical Sciences, New York Univ., 2005. Available at arXiv:math/0501141. | MR 2706830 | Zbl 1067.60099

[13] A. Sarkar and R. Sun. Brownian web in the scaling limit of supercritical oriented percolation in dimension 1+1. Electron. J. Probab. 18 (2013) Article 21. | MR 3035749 | Zbl 1290.60107

[14] R. Sun and J. M. Swart. The Brownian net. Ann. Probab. 36 (2008) 1153-1208. | MR 2408586 | Zbl 1143.82020

[15] B. Tóth and W. Werner. The true self-repelling motion. Probab. Theory Related Fields 111 (1998) 375-452. | MR 1640799 | Zbl 0912.60056