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Localisation et Concentration de la Marche de Sinai

Abstract : Sinai's walk is an elementary model of one dimensional random walk in random environment doing nearest neighbourhood jump. We impose three conditions on the random environment: two necessaries hypothesis to get a recurrent process but not a simple random walk and a hypothesis of regularity which allows us to have a good control on the fluctuations of the random environment. The asymptotic behaviour of such a walk was discovered by Y. Sinai in 1982: he shows that this process is sub-diffusive and that at time n it is located in the neighbourhood of a well defined point of the lattice. This point is a random variable depending only on the random environment and n, his explicit limit distribution was given by H. Kesten and A. O. Golosov (independently) in 1986. A part of this work (part II) gives an alternative proof of Sinai's results. The detailed study of the results of localization has motivated the study of a new aspect of the behaviour of Sinai' walk, we called it concentration phenomena (part III of the present thesis). We prove that it is concentrated in a small neighbourhood of the point of localization; this means that for an interval of time n Sinai's walk spends the quasi totality of this amount of time n in the neighbourhood of the point of localization. The size of this neighbourhood is negligible comparing to the typical range of Sinai's walk. The other result we show is that the local time of this random walk on the point of localization normalized by n converges in probability to a random variable depending only on n and on the random environment. This random variable is the inverse of the mean of the local time in the valley where the walk is prisoner, in a return time to the point of localization. All our results are “quenched” results, this mean that we work with a fixed environment that belongs to a probability subset of the random media and it is shown that this probability subset has a probability that goes to one. From these results we give some consequences on the maximum of the local time and the favourite site of Sinai's walk, in particular we show that all the favourite site and Sinai's walk, properly normalized, have the same limiting distribution.
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Submitted on : Thursday, January 8, 2004 - 12:19:48 PM
Last modification on : Thursday, September 13, 2018 - 12:08:03 PM
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Pierre Andreoletti. Localisation et Concentration de la Marche de Sinai. Analyse de données, Statistiques et Probabilités [physics.data-an]. Université de la Méditerranée - Aix-Marseille II, 2003. Français. ⟨tel-00004116⟩

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