Introduction

Abstract

Random matrix theory was introduced in statistics by Wishart [206] in the thirties, and then in theoretical physics by Wigner [205]. Since then, it has developed separately in a wide variety of mathematical fields, such as number theory, probability, operator algebras, convex analysis etc.

Therefore, lecture notes on random matrices can only focus on special aspects of the theory; for instance, the well-known book by Mehta [153] displays a detailed analysis of the classical matrix ensembles, and in particular of their eigenvalues and eigenvectors, the recent book by Bai and Silverstein [10] emphasizes the results related to sample covariance matrices, whereas the book by Hiai and Petz [117] concentrates on the applications of random matrices to free probability and operator algebras. The book in progress [6] in collaboration with Anderson and Zeitouni will try to take a broader and more elementary point of view, but still without relations to number theory or Riemann Hilbert approach for instance. The first of these topics is reviewed briefly in [126] and the second is described in [73].