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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. gallesco@ime.usp.br
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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

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 257 - 292
Copyright
© EDP Sciences, SMAI, 2013

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