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Theses

Etude de la diffusion des processus déterministes et faiblement aléatoires en environnement aléatoire

Abstract : This thesis studies the diffusion in the mirrors model, a physics-based model introduced in 1988 by Ruijgrok and Cohen. This model is deterministic and reversible. To treat this difficult model, initially defined only in dimension 2, we first generalized it to a model valid in any dimension. Initial numerical studies suggested that the model is diffusive in dimensions greater than or equal to 3. We then explored a perturbative diffusion coefficient approach based on the lace expansion technique developed by Gordon Slade for the study of self-avoiding random walk. Faced with the difficulty of the calculations, we slightly simplified the model by giving up the reversibility constraint. We thus obtained a new model that we call the permutations model. We then transformed these two models into random walks in random environment using a systematic and general approach. Thanks to these modifications, we were able to push the perturbative approach to obtain a satisfactory approximation of the value of the diffusion coefficient in the permutations model. The main result is the existence of a series in which all terms are well defined and the first terms provide the desired approximation. The convergence of this series remains an open problem. The analytical results are supported by a numerical approach to these models, which shows that the lace expansion gives quality results. Many questions remain open, including the calculation of the following terms of perturbative development and the generalization of this approach to the mirrors model -which should not be a problem- and then to a broader class of models.
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Yann Chiffaudel. Etude de la diffusion des processus déterministes et faiblement aléatoires en environnement aléatoire. Probabilités [math.PR]. Université Paris Cité, 2019. Français. ⟨NNT : 2019UNIP7083⟩. ⟨tel-03033731⟩

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