Strong and false asymptotic freedoms of large random matrices.

Summary This thesis is part of the theory of random matrices, at the intersection with the theory of free probabilities and operator algebras. It is part of a general approach that has proven itself in the last decades: importing techniques and concepts from non-commutative probability theory for the study of the spectrum of large random matrices. We are interested here in generalizations of the asymptotic freedom theorem of Voiculescu. In Chapters 1 and 2, we show strong asymptotic freedom results for Gaussian, unitary random and deterministic matrices. In Chapters 3 and 4, we introduce the notion of false asymptotic freedom for deterministic matrices and some Hermitian matrices with independent subdiagonal entries, interpolating the Wigner and Lévy matrix models.