UCLA

This thesis presents new results on spectral statistics of different families of large random matrices. Our main tool is certain types of {\textit{local estimates}} of the resolvents (or the Green's functions) of the random matrices, which are generally referred to as {\textit{local laws}}. Utilizing the standard approach developed over the last decade \cite{Yau_book} combined with a comparison method developed recently in \cite{Anisotropic}, we are able to prove (almost) optimal local laws for various random matrix ensembles with correlated and heavy-tailed entries. With these local laws, we establish the following three results.

We first study the largest eigenvalues for separable covariance matrices of the form $\mathcal Q :=A^{1/2}XBX^*A^{1/2}$. Here $X=(x_{ij})$ is an $n\times N$ random matrix, whose entries are $i.i.d.$ random variables with mean zero and variance $N^{-1}$; $A$ and $B$ are respectively $n \times n$ and $N\times N$ deterministic non-negative definite symmetric (or Hermitian) matrices. Under a sharp fourth moment tail condition, we prove that the limiting distribution of the largest eigenvalues of $\mathcal Q$ is universal under an $N^{2/3}$ scaling, as long as ${n}/{N}$ converges to a finite $d \in (0, \infty)$ as $N\to \infty$. In particular, if $B=I$, then $\mathcal Q$ becomes the sample covariance matrix, which is one of the most fundamental objects of study in high-dimensional statistics. Our result provides the strongest edge universality result for large dimensional sample covariance matrices so far.

Then we study the {\textit{eigenvector empirical spectral distribution}} (VESD)---an important tool in studying the limiting behavior of eigenvectors---for large separable covariance matrices. Under certain low moment assumptions, we prove an optimal convergence rate of the VESD to an anisotropic Mar{\v c}enko-Pastur law in the metric of Kolmogorov distance. Our results improve the suboptimal convergence rate in \cite{XYZ2013} under much more relaxed assumptions.

Finally, we study the eigenvalue distribution of a deformed non-Hermitian random matrix ensemble of the form $TX$, where $T$ is a deterministic $N\times M$ matrix and $X$ is a random $M\times N$ matrix with independent entries, each of which has zero mean and variance $(N\wedge M)^{-1}$. We prove the empirical spectral distribution (ESD) of $TX$ converges to an inhomogeneous local circular law, which is determined by the singular values of $T$. Moreover, the convergence holds up to the (almost) optimal local scale $(N\wedge M)^{-1/2+\epsilon}$ for any $\epsilon>0$. Our proof depends on a lower tail estimate for the smallest singular value of $TX-z$ for any $z\in \mathbb C$. This is also provided in this thesis.