The δn statistic for the β-Hermite ensemble

The fluctuation δn of the nth unfolded eigenvalue was recently characterized for the classical Gaussian ensembles of N×N random matrices (GOE, GUE, GSE). It is investigated here for the β-Hermite ensemble as a function of the reciprocal of the temperature β by Monte Carlo simulations. The ensemble-averaged fluctuation 〈δn2〉 and the autocorrelation function vary logarithmically with n for any β>0 (1≪n≪N). The simple logarithmic behavior of the higher-order moments of δn, reported in the literature for the GOE (β=1) and the GUE (β=2), holds for any β>0 and is accounted for by Gaussian distributions whose variances depend linearly on lnn.