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Comportements asymptotiques et transition de phase pour des marches aléatoires en milieux aléatoires et des marches renforcées

Abstract : In this thesis, we study some behaviours of random walks in random environemnts and reinforced random walks. We will first look at random walks in Dirichlet environment and then at two models of reinforced walks: the linealry edge-reinforced random walk and the vertex reinforced jump process. Random walks in Dirichlet environment are a special case of random walk in random environments that exhibit an important property simplifiant leur étude: the statistical invariance by time reversal. We will first use this property to characterize the asymptotic behaviour of these walks in dimensions 3 and higher when they are transient with zero speed. In this case we show that their behaviour is characterized by a stable process. Then we show that this property of statistical invariance by time reversal is actually characteristic of random walks in Dirichlet environments. The linearly edge-reinforced random walk and the vertex reinforced jump process are two closely linked models of reinforced processes. In both models the walk tends to come back to areas it has already visited. We will first show that some characteristic quantities of the model exhibit some monotonicity in their parameters. This induces some consequences: unicity for the phase transition between recurrence and transience, recurrence in dimension 2, and a 0-1 law for recurrence. Then, we will look at a biased version of the linearly edge-reinforced random walk for which we show that its behaviour stays similar to the original model on some infinite graphs
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Rémy Poudevigne-Auboiron. Comportements asymptotiques et transition de phase pour des marches aléatoires en milieux aléatoires et des marches renforcées. Probabilités [math.PR]. Université de Lyon, 2020. Français. ⟨NNT : 2020LYSE1128⟩. ⟨tel-03300734⟩

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