Spectral statistics of non-Hermitian and non-linear random matrix models

Abstract

This thesis presents several results on the universality phenomena of large random matrices. The first part of this thesis concerns the ensemble of non-Hermitian random matrices with independent identically distributed (i.i.d.) entries. We study the behavior of the eigenvalues and eigenvectors of this ensemble in two symmetry classes: with real-valued and complex-valued entries. Firstly, we consider an ensemble of real non-symmetric i.i.d. matrices with entries that have finite moments. We show that its $k$-point correlation function in the bulk away from the real line converges to a universal limit. This work builds on the previous result of Maltsev and Osman \cite{MO}, showing the local bulk universality in the complex for the complex version of this ensemble. Secondly, we consider a constant-size subset of left and right eigenvectors of an $N\times N$ i.i.d. complex non-Hermitian matrix associated with the eigenvalues with pairwise distances at least $N^{-\frac12+\epsilon}$. We show that arbitrary constant rank projections of these eigenvectors are asymptotically Gaussian and jointly independent. Finally, motivated by applications in statistics, we consider certain large random matrices, called \emph{random inner-product kernel matrices}, which are essentially given by a non-linear function $f$ applied entrywise to a sample-covariance matrix, $f(X^TX)$, where $X \in \R^{d \times N}$ is random and normalized in such a way that $f$ typically has order-one arguments. We consider the \emph{polynomial regime}, where $N \asymp d^\ell$ for some $\ell > 0$. Earlier work by various authors showed that, when the columns of $X$ are either uniform on the sphere or standard Gaussian vectors, and when $\ell$ is an integer (the linear regime $\ell = 1$ is particularly well-studied), the bulk eigenvalues of such matrices behave in a simple way: They are asymptotically given by the free convolution of the semicircular and Mar\v{c}enko--Pastur distributions, with relative weights given by expanding $f$ in the Hermite basis. In the final part of this thesis, we show that this phenomenon is universal, holding as soon as $X$ has i.i.d. entries with all finite moments.