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marches aleatoires en milieu aleatoire et marches branchantes

Abstract : This thesis deals with two models of random walks. The first model belongs to the family of random walks in random environment. In the case where the graph is a Galton-watson tree, we are interested in the asymptotic properties of the walk. When the walk is transient, we look at its speed. We obtain an explicit criterion to have a positive speed, and we give the order of magnitude of the distance to the root when the speed is zero. We apply these results to show that the linearly edge reinforced random walk on a regular tree always has positive speed. Then we study speed-up and slowdown probabilities of the walk and give a large deviations principle in the quenched and annealed settings. Under the annealed probability, we distinguish several regimes of large deviations. The second part of the work presents a model of branching random walks with absorption. We model the evolution of a population moving on the positive half-line of the real axis, in which particles die when crossing zero. Two regimes exist depending on whether the population survives forever or not. In the case where the population dies out, we find the asymptotic equivalents of the survival probability at time n.
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Contributor : Elie Aidekon <>
Submitted on : Wednesday, October 28, 2009 - 3:19:37 PM
Last modification on : Wednesday, December 9, 2020 - 3:08:56 PM
Long-term archiving on: : Thursday, June 17, 2010 - 6:31:59 PM


  • HAL Id : tel-00426925, version 1


Elie Aidekon. marches aleatoires en milieu aleatoire et marches branchantes. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2009. Français. ⟨tel-00426925⟩



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