Nearest-neighbour spacing distributions of the β-Hermite ensemble of random matrices

Abstract

The evolution with β of the distributions of the spacing ‘ s’ between nearest-neighbour levels of unfolded spectra of random matrices from the β-Hermite ensemble ( β-HE) is investigated by Monte Carlo simulations. The random matrices from the β-HE are real symmetric and tridiagonal where β, which can take any positive value, is the reciprocal of the temperature in the classical electrostatic interpretation of eigenvalues. The distribution of eigenvalues coincide with those of the three classical Gaussian ensembles for β=1, 2, 4. The use of the β-HE ensemble results in an incomparable speed up and efficiency of numerical simulations of all spectral characteristics of large random matrices. Generalized gamma distributions are shown to be excellent approximations of the nearest-neighbor spacing (NNS) distributions for any β while being still simple. They account both for the level repulsion in ∼ s β when s→0 and for the whole shape of the NNS distributions in the range of ‘ s’ which is accessible to experiment or to most numerical simulations. The exact NNS distribution of the GOE ( β=1) is in particular significantly better described by a generalized gamma distribution than it is by the Wigner surmise while the best generalized gamma approximation coincides essentially with the Wigner surmise for β>∼2. They describe too the evolution of the level repulsion between that of a Poisson distribution and that of a GOE distribution when β increases from 0 to 1. The distribution of ln ( s), related to the electrostatic interaction energy between neighbouring charges, is accordingly well approximated by a generalized Gumbel distribution for any β⩾0. The distributions of the minimum NN spacing between eigenvalues of matrices from the β-HE, obtained both from as-calculated eigenvalues and from unfolded eigenvalues are Brody distributions which are classically used to characterize the spectral fluctuations of various physical systems.