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, Volume 46 (2010) no. 4, p. 991-1000

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.

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.

DOI : https://doi.org/10.1214/09-AIHP336
Classification:  60K37,  60F05
Keywords: 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},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {4},
     year = {2010},
     pages = {991-1000},
     doi = {10.1214/09-AIHP336},
     zbl = {1210.82028},
     mrnumber = {2744882},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2010__46_4_991_0}
}
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, Volume 46 (2010) no. 4, pp. 991-1000. doi : 10.1214/09-AIHP336. http://www.numdam.org/item/AIHPB_2010__46_4_991_0/

[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.