Random Matrices and Random Partitions ― Normal Convergence

This book is aimed at graduate students and researchers who are interested in the probability limit theory of random matrices and random partitions. It mainly consists of three parts. Part I is a brief review of classical central limit theorems for sums of independent random variables, martingale sequences and Markov chains, etc. These classical theorems are frequently used in the study of random matrices and random partitions where random matrices are well-studied in probability theory. Part II concentrates on the asymptotic distribution theory of Circular Unitary Ensemble and Gaussian Unitary Ensemble, which are prototypes of random matrix theory. It turns out that the classical central limit theorems and methods are applicable in describing asymptotic distributions of eigenvalue statistics like linear functionals of eigenvalues. This is attributed to the nice algebraic structures of models. This part also studies the Circular s Ensembles and Gaussian s Ensembles, which may be viewed as extensions of the Circular Unitary Ensemble and Gaussian Unitary Ensemble. Part III is devoted to the study of random uniform and Plancherel partitions. As is known, there is a surprising similarity between random matrices and random integer partitions from the viewpoint of asymptotic distribution theory, though it is difficult to find any direct link between the two finite models.This book treats only second-order fluctuations for primary random variables from two classes of special random models. It is written in a clear, concise and pedagogical way. It may be read as an introductory text to further study probability theory of general random matrices, random partitions and even random point processes. This book is aimed at graduate students and researchers who are interested in probability limit theory of random matrices and random integer partitions.