The Arcsine law as the limit of the internal DLA cluster generated by Sinai’s walk

N. Enriquez; C. Lucas; F. Simenhaus

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 4, page 991-1000
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Enriquez, N., Lucas, C., and 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 46.4 (2010): 991-1000. <http://eudml.org/doc/241847>.

@article{Enriquez2010,
abstract = {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.},
author = {Enriquez, N., Lucas, C., Simenhaus, F.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Sinai’s walk; internal DLA; random walks in random environments; excursion theory; Sinai's walk},
language = {eng},
number = {4},
pages = {991-1000},
publisher = {Gauthier-Villars},
title = {The Arcsine law as the limit of the internal DLA cluster generated by Sinai’s walk},
url = {http://eudml.org/doc/241847},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Enriquez, N.
AU - Lucas, C.
AU - Simenhaus, F.
TI - The Arcsine law as the limit of the internal DLA cluster generated by Sinai’s walk
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 4
SP - 991
EP - 1000
AB - 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.
LA - eng
KW - Sinai’s walk; internal DLA; random walks in random environments; excursion theory; Sinai's walk
UR - http://eudml.org/doc/241847
ER -

References

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  1. [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. Zbl0955.60046MR1746300
  2. [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. Zbl0776.60128MR1218674
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  4. [4] G. F. Lawler, M. Bramson and D. Griffeath. Internal diffusion limited aggregation. Ann. Probab. 20 (1992) 2117–2140. Zbl0762.60096MR1188055
  5. [5] P. Lévy. Sur certains processus stochastiques homogènes. Compos. Math. 7 (1939) 283–339. Zbl0022.05903MR919
  6. [6] R. Mansuy and M. Yor. Aspects of Brownian Motion. Springer, Berlin, 2008. Zbl1162.60022MR2454984
  7. [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. Zbl0917.60006MR1725357
  8. [8] Y. G. Sinai. The limit behavior of a one-dimensional random walk in a random environment. Teor. Veroyatn. Primen. 27 (1982) 247–258. Zbl0497.60065MR657919
  9. [9] S. Tavaré and O. Zeitouni. Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1837. Springer, Berlin, 2004. Zbl1034.60001MR2071629
  10. [10] M. Yor. Local Times and Excursions for Brownian Motion: A Concise Introduction Lecciones en Mathematicas. Universidad Central de Venezuela, Caracas, 1995. 

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