The Arcsine law as the limit of the internal DLA cluster generated by Sinai's walk
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 4, pp. 991-1000.

On détermine la loi limite du cluster de diffusion à agrégation limitée interne comme celle d'une fonctionnelle du mouvement brownien, qui donne une nouvelle interprétation de la loi de l'Arcsinus.

We identify the limit of the internal DLA cluster generated by Sinai's walk as the law of a functional of a brownian motion which turns out to be a new interpretation of the Arcsine law.

DOI : https://doi.org/10.1214/09-AIHP336
Classification : 60K37,  60F05
Mots clés : Sinai's walk, internal DLA, random walks in random environments, excursion theory
@article{AIHPB_2010__46_4_991_0,
     author = {Enriquez, N. and Lucas, C. and Simenhaus, F.},
     title = {The Arcsine law as the limit of the internal DLA cluster generated by Sinai's walk},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {991--1000},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {4},
     year = {2010},
     doi = {10.1214/09-AIHP336},
     zbl = {1210.82028},
     mrnumber = {2744882},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/09-AIHP336/}
}
Enriquez, N.; Lucas, C.; Simenhaus, F. The Arcsine law as the limit of the internal DLA cluster generated by Sinai's walk. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 4, pp. 991-1000. doi : 10.1214/09-AIHP336. http://www.numdam.org/articles/10.1214/09-AIHP336/

[1] J. Bertoin. Subordinators: Examples and applications. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997). Lecture Notes in Math. 1717 1-91. Springer, Berlin, 1999. | MR 1746300 | Zbl 0955.60046

[2] P. Diaconis and W. Fulton. A growth model, a game, an algebra, Lagrange inversion, and characteristic classes. Rend. Sem. Mat. Univ. Politec. Torino 49 (1993) 95-119. | MR 1218674 | Zbl 0776.60128

[3] P. G. Hoel, S. C. Port and C. J. Stone. Introduction to Stochastic Processes. Houghton Mifflin, Boston, MA, 1972. | MR 358879 | Zbl 0258.60003

[4] G. F. Lawler, M. Bramson and D. Griffeath. Internal diffusion limited aggregation. Ann. Probab. 20 (1992) 2117-2140. | MR 1188055 | Zbl 0762.60096

[5] P. Lévy. Sur certains processus stochastiques homogènes. Compos. Math. 7 (1939) 283-339. | Numdam | MR 919 | Zbl 0022.05903

[6] R. Mansuy and M. Yor. Aspects of Brownian Motion. Springer, Berlin, 2008. | MR 2454984 | Zbl 1162.60022

[7] D. Revuz and M. Yor. Continuous martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin, 1999. | MR 1725357 | Zbl 0917.60006

[8] Y. G. Sinai. The limit behavior of a one-dimensional random walk in a random environment. Teor. Veroyatn. Primen. 27 (1982) 247-258. | MR 657919 | Zbl 0497.60065

[9] S. Tavaré and O. Zeitouni. Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1837. Springer, Berlin, 2004. | MR 2071629 | Zbl 1034.60001

[10] M. Yor. Local Times and Excursions for Brownian Motion: A Concise Introduction Lecciones en Mathematicas. Universidad Central de Venezuela, Caracas, 1995.