ON PROFESSOR BAI'S MAIN CONTRIBUTIONS TO THE SPECTRAL THEORY ON RANDOM MATRICES

Abstract

The aim of the spectral theory of large dimensional random matrices (RMT) is to investigate the limiting behaviour of the eigenvalues (A,,j) of a random matrix (A,) when its sizes tend to infinity. Of particular interest are the empirical spectral distribution (ESD) F, := n-l Cj d ~ , , ~ , the extreme eigenvalues Amax(An) = maxj An,j and Xmin(An) = minj A,,j, or the spacings {A,,j A,,j-l}. The main underlying mathematical problems for a given class of random matrices (A,) are the following: a) find a limiting spectral distribution (LSD) G to which converges the sequence of ESD (F,) ; b) find the limits of the extreme eigenvalues Amax(A,) and Amin(An) ; c) quantify the rates of the convergences a) and b). d) find second order limit therems such as central limit theorems for the convergences a) and b). Professor Bai, one of the world leading experts of the field, has brought several major contributions to the theory.