Eigenvalue distribution of large random matrices, from one matrix to several coupled matrices

Abstract

It has been observed that the statistical distribution of the eigenvalues of random matrices possesses universal properties, independent of the probability law of the stochastic matrix. In this article we find the correlation functions of this distribution in two classes of random Hermitian matrix models: the one-matrix model, and the two-matrix model, although it seems that the methods and conclusions presented here will allow generalization to other multi-matrix models such as the chain of matrices, or the O( n) model. We recover the universality of the two-point function in two regimes: the short distance regime when the two eigenvalues are separated by a small number of other eigenvalues, and on the other hand the long range regime, when the two eigenvalues are far away in the spectrum. In this regime we have to smooth the short scale oscillations. We also discuss the universality properties of more than two eigenvalues correlation functions.