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Abstract: Free probability is a part of quantum probability, which stems in the 80's from a new concept of independence due to D. Voiculescu, called freeness. In our thesis, we deal with this theory from several points of view. We first focuse our attention to conditional freeness, which is a generalization of freeness introduced by R. Speicher and M. Bozejko. We describe it in a combinatorial way similar to the one realized by G.-C. Rota for classical independence and R. Speicher for freeness. We therefore have to define a new class of non-crossing partitions, which links togethers classical non-crossing partitions and interval -- or boolean -- partitions. This work produces a simple approach to conditionally free convolution. We define next a new notion of freeness, which can be charactherized as a conditional freeness with respect to a sequence of states. Then, in a radical different way, we study the asymptotic behaviour of the empirical law of some random matrices when their dimension grows to infinity. We use matricial and free stochastic calculus. We follow there a way opened by D. Voiculescu when he proved that gaussian independent matrices were asymptotically free. We are able to establish some fluctuations and large deviations results -- in a joined work with A. Guionnet -- for Wigner matrices, Wishart matrices and a couple of independent matrices. We are led to define new concepts of free entropy and free information, and to prove inequalities between them in an analogous way to the one D. Bakry and M. Emery produced for log-Sobolev inequalities. Our new definition of free entropy is very close to Voiculescu's second version using a non-commutative Hilbert transform.
keywords: Free Probability, Random Matrices, Large
Deviations.
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