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Theses

Random walks in random environment on Z:
localization studies in the recurrent and transient cases

Abstract : Random walks in random environment is a
suitable model for diffusion and transport in inhomogeneous media
that have regularity properties on a macroscopic scale. The first
introductive chapter illustrates the wide variety of behaviors that
are captured by the random walks in random environment model. The
second chapter concerns Sinai's walk (the recurrent case), which is
known for a phenomenon of strong localization. Our main result shows
a weakness of this localization phenomenon. In particular, we give a
negative answer to a problem of Erdös and Révész, originally
formulated for the usual homogeneous random walk. In the third
chapter, we focus our attention on the upper limits of Sinai's walk
in random scenery and treat a conjecture of Révész. The fourth
and fifth chapters deal with transient random walks in random
environment with zero asymptotic speed. A classical result of
Kesten, Kozlov and Spitzer says that the hitting time of the level
n converges in law, after a proper normalization, towards a
positive stable law, but they do not obtain a description of its
parameter. A different proof of this result is presented: a close
study of the potential associated to the environment leads to a
complete characterization of this stable law. The case of Dirichlet
environment turns out to be remarkably explicit.
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https://tel.archives-ouvertes.fr/tel-00158859
Contributor : Olivier Zindy <>
Submitted on : Friday, June 29, 2007 - 8:43:10 PM
Last modification on : Wednesday, December 9, 2020 - 3:10:15 PM
Long-term archiving on: : Thursday, April 8, 2010 - 10:09:56 PM

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  • HAL Id : tel-00158859, version 1

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Olivier Zindy. Random walks in random environment on Z:
localization studies in the recurrent and transient cases. Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2007. English. ⟨tel-00158859⟩

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