General tridiagonal random matrix models, limiting distributions and fluctuations

Abstract

In this paper we discuss general tridiagonal matrix models which are natural extensions of the ones given in Dumitriu and Edelman (J. Math. Phys. 43(11): 5830-5847, 2002; J. Math. Phys. 47(11):5830-5847, 2006). We prove here the convergence of the distribution of the eigenvalues and compute the limiting distributions in some particular cases. We also discuss the limit of fluctuations, which, in a general context, turn out to be Gaussian. For the case of several random matrices, we prove the convergence of the joint moments and the convergence of the fluctuations to a Gaussian family. The methods involved are based on an elementary result on sequences of real numbers and a judicious counting of levels of paths. © 2008 Springer-Verlag.