Numerical Methods for Eigenvalue Distributions of Random Matrices

We present efficient numerical techniques for calculation of eigenvalue distributions of random matrices in the beta-ensembles. We compute histograms using direct simulations on very large matrices, by using tridiagonal matrices with appropriate simplifications. The distributions are also obtained by numerical solution of the Painlevé II and V equations with high accuracy. For the spacings we show a technique based on the Prolate matrix and Richardson extrapolation, and we compare the distributions with the zeros of the Riemann zeta function. 1 Largest Eigenvalue Distributions In this section, the distributions of the largest eigenvalue of matrices in the β-ensembles are studied. Histograms are created first by simulation, then by solving the Painlevé II nonlinear differential equation. 1.1 Simulation The Gaussian Unitary Ensemble (GUE) is defined as the Hermitian n × n matrices A, where the diagonal elements xjj and the upper triangular elements xjk = ujk + ivjk are independent Gaussians with zero-mean, and