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Grandes déviations et convergence du spectre de matrices aléatoires

Abstract : One of the main objects of random matrix theory is the spectrum of matrices of large dimension and whose entries are random variables. In this thesis, we concern ourselves with two questions : the large deviations of the largest eigenvalue and the convergence of the empirical measure. Apart from the case of coefficients with Gaussian or heavy-tailed distributions., few large deviation principles are proved in random matrix theory, for the empirical measure or the largest eigenvalue. In the first part of this thesis, we prove a series of large deviation principles for the largest eigenvalue of Wigner matrices, Wishart matrices and matrices with variance profiles with coefficients whose distributions satisfy a sub-Gaussian bound.In the second part of this thesis, we examine the question of the convergence of the empirical measure of polynomials of random matrices. In the case of self-adjoint polynomials in independent Wigner matrices, thanks to the results of Voiculescu, we can see the limit measure as the spectral measure of a polynomial in independently free semi-circular elements. But in the general case, it is necessary to control the smallest singular values in order to conclude. We present such a control for polynomials of degree 2 in Ginibre matrices and we prove the convergence of the empirical measure.
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Jonathan Husson. Grandes déviations et convergence du spectre de matrices aléatoires. Probabilités [math.PR]. Université de Lyon, 2019. Français. ⟨NNT : 2019LYSEN067⟩. ⟨tel-02497186⟩



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