Random Matrix Theory and Its Applications

This article reviews the important ideas behind random matrix theory (RMT), which has become a major tool in a variety of disciplines, including mathematical physics, number theory, combinatorics and multivariate statistical analysis. Much of the theory involves ensembles of random matrices that are governed by some probability distribution. Examples include Gaussian ensembles and Wishart–Laguerre ensembles. Interest has centered on studying the spectrum of random matrices, especially the extreme eigenvalues, suitably normalized, for a single Wishart matrix and for two Wishart matrices, for finite and infinite sample sizes in the real and complex cases. The Tracy–Widom Laws for the probability distribution of a normalized largest eigenvalue of a random matrix have become very prominent in RMT. Limiting probability distributions of eigenvalues of a certain random matrix lead to Wigner’s Semicircle Law and Marc˘enko–Pastur’s Quarter-Circle Law. Several applications of these results in RMT are described in this article.