The Distribution Of Eigenvalues Of Large Dimensional Random Matrices Generated From Four Manifolds

In recent decades, random matrix theory has been found valuable in an increasing number of disciplines of science and engineering, including Mathematics, Statistics, Physics, Communication, biology, etc. And new results in random matrix theory and its applications are being established almost everyday. The flexibility, universality, predictability as well as the rich mathematical structure of the random matrices attract a group researchers from Mathematics and other diverse fields. And the theoretic research in the hot area of multiple disciplines converging is becoming more mature.One of the major research interests in random matrix theory is the statistical properties of their eigenvalues and eigenvectors. From the classical Wishart matrix ensemble (emerging from data analysis) to the Wigner matrix ensemble (applicable to heavy-nuclei atoms), random matrix theory is developing and motivated with the requirements of various disciplines’applications. During the study processes, many universal laws (i.e. the laws are independent of matrix entries’explicit distributions but only based on the matrix structures) are discovered, such as Winger’s semicircle law, Marcenko-Pastur law, as well as Girko’s circular law. There are also other universal local spectral distributions, like Tracy-Widom distributions, sine kernel distribution, which are playing a role in random matrix theory just like the role of normal distribution in classical central limit theorem in some sense. These local spectral distributions are not necessarily based on specific matrix ensembles, and they are found ubiquitous in other Mathematical and Physical models beyond random matrices.In this thesis, we will study three kinds of random matrix models which are "sample-covariance" type matrices, Euclidean random matrices and random inner-product matrices. They are found very useful in multivariate statistics and statistical physics, and the research in the latter two types of high dimensions are just at their beginning. Different from the matrix models in classical random matrix theory, what we study are those matrices with dependent entries in different cases. The three types of matrix ensembles we just mentioned are mainly generated from the lp norm uniformly distributed points in four regular manifolds:unit lp hyper-ball, unit lp hyper-sphere, lp hyper-ellipsoid and its surface. And for the random inner-product kernel matrices, we also analyze a general setting when the random vectors have isotropic and log-concave distributions and prove a conjecture recently proposed by Do and Vu.The empirical distributions of eigenvalues of random matrices is one of the major research interests in random matrix theory, and it is also the premise and foundation for the study of other spectral properties. For the three random matrix ensembles generated from the above several samples, we investigate their empirical spectral distributions under two kinds of high dimensional settings, and obtain separately their explicit limits. Precisely, when sample numbers n tends to infinity, and n/N (N is the dimension of samples) converges to a non-zero constant, we derive the limiting spectral distributions for Gram matrices (of the columns), large Euclidean random matrices and random inner-product kernel matrices which are generated from different random vectors. And when n/Nâ†'O (which is also of great statistical significance), through re-scaling the sample covariance matrices, Euclidean random matrices and random inner-product kernel matrices, we have limiting spectral distributions related to the celebrated Wigner’s semicircle law.The approach in this paper is to connect our matrix models to the classical random matrix ensembles. By applying the probability theory and classical random matrix theory, we obtain our main results. These results are derived by a large numbers of tedious estimations on the random vectors generated from the four manifolds. For the Euclidean random matrices and random inner-product kernel matrices, we decompose them into several independent random matrices based on the Taylor expansion. And then using the statistical properties of the random vectors when lp-norm uniformly sampled from four manifolds or the collection of all N dimensional isotropic and log-concave vectors, we establish the corresponding conclusions based on some classical results in random matrix theory.