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Théorie des grandes déviations en physique statistique : quelques aspects théoriques et numériques

Abstract : This thesis is concerned with various aspects of large deviations theory in relation with statistical physics. Both theoretical and numerical considerations are dealt with. The first part of the work studies long time large deviations properties of diffusion processes. First, we prove new ergodicity results for Feynman-Kac dynamics, both in continuous and discrete time. This leads to new fine results (in the sense of topology) for large deviations of empirical measures of diffusion processes. Various numerical problems are then covered. We first provide precise error estimates on discretizations of Feynman-Kac dynamics, for which the nonlinear features of the dynamics demand new tools. In order to reduce the variance of naive estimators, we provide an adaptive algorithm relying on the technique of stochastic approximation. We finally consider a problem concerning low temperature systems. We present a new method for constructing an approximation of the optimal control from the instanton (or reaction path) theory. The last part of the thesis is concerned with the different topic of Coulomb gases, which appear both in physics and random matrix theory. We first present an efficient method for simulating such gases, before turning to gases under constraint. For such gases, we prove new concentration results in the limit of a large number of particles, under some conditions on the constraint. We also present a simulation algorithm, which confirms the theoretical expectations
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Submitted on : Monday, June 29, 2020 - 5:31:09 PM
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Grégoire Ferré. Théorie des grandes déviations en physique statistique : quelques aspects théoriques et numériques. Statistiques [math.ST]. Université Paris-Est, 2019. Français. ⟨NNT : 2019PESC1035⟩. ⟨tel-02884212⟩



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