Isolated eigenvalues of non Hermitian random matrices

Abstract : This thesis is about spiked models of non Hermitian random matrices. More specifically, we consider matrices of the type A+P, where the rank of P stays bounded as the dimension goes to infinity and where the matrix A is a non Hermitian random matrix. We first prove that if P has some eigenvalues outside the bulk, then A+P has some eigenvalues (called outliers) away from the bulk. Then, we study the fluctuations of the outliers of A around their limit and prove that they are distributed as the eigenvalues of some finite dimensional random matrices. Such facts had already been noticed for Hermitian models. More surprising facts are that outliers can here have very various rates of convergence to their limits (depending on the Jordan Canonical Form of P) and that some correlations can appear between outliers at a macroscopic distance from each other. The first non Hermitian model studied comes from the Single Ring Theorem due to Guionnet, Krishnapur and Zeitouni. Then we investigated spiked models for nearly Hermitian random matrices : where A is Hermitian but P isn’t. At last, we studied the outliers of Gaussian Elliptic random matrices. This thesis also investigates the convergence in distribution of random variables of the type Tr( f (A)M) where A is a matrix from the Single Ring Theorem and f is analytic on a neighborhood of the bulk and the Frobenius norm of M has order √N. As corollaries, we obtain central limit theorems for linear spectral statistics of A (for analytic test functions) and for finite rank projections of f (A) (like matrix entries).
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Jean Rochet. Isolated eigenvalues of non Hermitian random matrices. General Mathematics [math.GM]. Université Sorbonne Paris Cité, 2016. English. ⟨NNT : 2016USPCB030⟩. ⟨tel-01589219⟩

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