Sturm sequences and the eigenvalue distribution of the beta-Hermite random matrix ensemble

Abstract

This paper proposes that the study of Sturm sequences is invaluable in the numerical computation and theoretical derivation of eigenvalue distributions of random matrix ensembles. We first explore the use of Sturm sequences to efficiently compute histograms of eigenvalues for symmetric tridiagonal matrices and apply these ideas to random matrix ensembles such as the [beta]-Hermite ensemble. Using our techniques, we reduce the time to compute a histogram of the eigenvalues of such a matrix from O(n2 + m) to O(mn) time where n is the dimension of the matrix and m is the number of bins (with arbitrary bin centers and widths) desired in the histogram. Our algorithm is a significant improvement because m is usually much smaller than n. This algorithm allows us to compute histograms that were computationally infeasible before, such as those for n equal to 1 billion. Second, we give a derivation of the eigenvalue distribution for the [beta]-Hermite random matrix ensemble (for general [beta]). The novelty of the approach presented in this paper is in the use of Sturm sequences to derive the distribution. We derive an analytic formula in terms of multivariate integrals for the eigenvalue distribution and the largest eigenvalue distribution for general [beta] by analyzing the Sturm sequence of the tridiagonal matrix model. Finally, we explore the relationship between the Sturm sequence of a random matrix and its shooting eigenvectors. We show using Sturm sequences that, assuming the eigenvector contains no zeros, the number of sign changes in a shooting eigenvector of parameter A is equal to the number of eigenvalues greater than [lambda].