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Dynamics of eigenvectors of random matrices and eigenvalues of nonlinear models of matrices

Abstract : This thesis consists in two independent parts. The first part pertains to the study of eigenvectors of random matrices of Wigner-type. Firstly, we analyze the distribution of eigenvectors of deformed Wigner matrices which consist in a perturbation of a Wigner matrix by a deterministic diagonal matrix. If the two matrices are of the same order of magnitude, it was proved that eigenvectors are completely delocalized and eigenvalues belongs to the Wigner-Dyson-Mehta universality class. We study here an intermediary phase where the deterministic perturbation dominates the randomness of the Wigner matrix : eigenvectors are not completely delocalized but eigenvalues are still universal. The eigenvector entries are asymptotically Gaussian with a variance which localize them onto an explicit part of the spectrum. Moreover, their mass is concentrated around their variance in a sense of a quantum unique ergodicity property. Then, we consider correlations of different eigenvectors. To do so, we exhibit a new observable on eigenvector moments of the Dyson Brownian motion. It follows a closed parabolic equation which is a fermionic counterpart of the Bourgade-Yau eigenvector moment flow. By combining the study of these two observables, it becomes possible to study some eigenvector correlations.The second part concerns the study of eigenvalue distribution of nonlinear models of random matrices. These models appear in the study of random neural networks and correspond to a nonlinear version of sample covariance matrices in the sense that a nonlinear function, called the activation function, is applied entrywise to the matrix. The empirical eigenvalue distribution converges to a deterministic distribution characterized by a self-consistent equation of degree 4 followed by its Stieltjes transform. The distribution depends on the function only through two explicit parameters. For a specific choice of these parameters, we recover the Marchenko-Pastur distribution which stays stable after going through several layers of the network.
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Submitted on : Wednesday, October 30, 2019 - 10:16:12 AM
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Lucas Benigni. Dynamics of eigenvectors of random matrices and eigenvalues of nonlinear models of matrices. Statistics [math.ST]. Université Sorbonne Paris Cité, 2019. English. ⟨NNT : 2019USPCC003⟩. ⟨tel-02338795⟩

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