Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-22T09:05:38.871Z Has data issue: false hasContentIssue false
  • English
  • Français

Meeting time of independent random walks in random environment

Published online by Cambridge University Press:  08 February 2013

Christophe Gallesco
Christophe Gallesco*
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, rua do Matão 1010, CEP 05508–090, São Paulo, SP, Brazil. [email protected]
Get access

Abstract

We consider, in the continuous time version, γ independent random walks on Z+ in random environment in Sinai’s regime. Let Tγ be the first meeting time of one pair of the γ random walks starting at different positions. We first show that the tail of the quenched distribution of Tγ, after a suitable rescaling, converges in probability, to some functional of the Brownian motion. Then we compute the law of this functional. Eventually, we obtain results about the moments of this meeting time. Being Eω the quenched expectation, we show that, for almost all environments ω, Eω[Tγc] is finite for c < γ(γ − 1) / 2 and infinite for c > γ(γ − 1) / 2.

Keywords

Random walk in random environmentSinai’s regimet-stable pointmeeting timecoalescing time
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 257 - 292
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Belitsky, V., Ferrari, P., Menshikov, M. and Popov, S., A mixture of the exclusion process and the voter model. Bernoulli 7 (2001) 119–144. Google Scholar
Böhm, W. and Mohanty, S.G., On the Karlin–McGregor theorem and applications. Ann. Appl. Probab. 7 (1997) 314–325. Google Scholar
Comets, F. and Popov, S.Yu., Limit law for transition probabilities and moderate deviations for Sinai’s random walk in random environment. Probab. Theory Relat. Fields 126 (2003) 571–609. Google Scholar
Comets, F. and Popov, S.Yu., A note on quenched moderate deviations for Sinai’s random walk in random environment. ESAIM : PS 8 (2004) 56–65. Google Scholar
Comets, F., Menshikov, M.V. and Popov, S.Yu., Lyapunov functions for random walks and strings in random environment. Ann. Probab. 26 (1998) 1433–1445. Google Scholar
Dembo, A., Gantert, N., Peres, Y. and Shi, Z., Valleys and the maximal local time for random walk in random environment. Probab. Theory Relat. Fields 137 (2007) 443–473. Google Scholar
Enriquez, N., Sabot, C. and Zindy, O., Aging and quenched localization one-dimensional random walks in random environment in the bub-ballistic regime. Bulletin de la S.M.F. 137 (2009) 423–452. Google Scholar
Fribergh, A., Gantert, N. and Popov, S.Yu., On slowdown and speedup of transient random walks in random environment. Probab. Theory Relat. Fields 147 (2010) 43–88. Google Scholar
C. Gallesco, On the moments of the meeting time of independent random walks in random environment. arXiv:0903.4697 (2009).
Gantert, N., Peres, Y. and Shi, Z., The infinite valley for a recurrent random walk in random environment. Ann. Inst. Henri Poincaré 46 (2010) 525–536. Google Scholar
Greven, A. and den Hollander, F., Large deviations for a random walk in random environment. Ann. Probab. 22 (1994) 1381 − 1428. Google Scholar
Hu, Y. and Shi, Z., Moderate deviations for diffusions with Brownian potentials. Ann. Probab. 32 (2004) 3191–3220. Google Scholar
B. Hughes, Random Walks and Random Environments. The Clarendon Press, Oxford University Press, New York. Random Environments 2 (1996).
Kesten, H., M.V.Kozlov and F. Spitzer, A limit law for random walk in a random environment. Compos. Math. 30 (1975) 145–168. Google Scholar
Komlós, J., Major, P. and Tusnády, G., An approximation of partial sums of independent RV’s and the sample DF. I. Z. Wahrscheinlichkeitstheor. Verw. Gebiete 32 (1975) 111–131. Google Scholar
Saloff-Coste, L., Lectures on Finite Markov Chains. Lectures on probability theory and statistics, Saint-Flour, 1996, Springer, Berlin. Lect. Notes Math. 1665 (1997) 301–413. Google Scholar
Z. Shi, Sinai’s Walk via Stochastic Calculus, in Milieux Aléatoires Panoramas et Synthèses 12, edited by F. Comets and E. Pardoux. Société Mathématique de France, Paris (2001).
Sinai, Ya.G., The limiting behavior of one-dimensional random walk in random medium. Theory Probab. Appl. 27 (1982) 256–268. Google Scholar
Solomon, F., Random walks in a random environment. Ann. Probab. 3 (1975) 1–31. Google Scholar
Zeitouni, O., Lecture Notes on Random Walks in Random Environment given at the 31st Probability Summer School in Saint-Flour, Springer. Lect. Notes Math. 1837 (2004) 191–312.Google Scholar