Accuracy and precision of the estimation of the number of missing levels in chaotic spectra using long-range correlations

Abstract

We study the accuracy and precision for estimating the fraction of observed levels $\varphi$ in quantum chaotic spectra through long-range correlations. We focus on the main statistics where theoretical formulas for the fraction of missing levels have been derived, the $\Delta_3$ of Dyson and Mehta and the power spectrum of the $\delta_n$ statistic. We use Monte Carlo simulations of the spectra from the diagonalization of Gaussian Orthogonal Ensemble matrices with a definite number of levels randomly taken out to fit the formulas and calculate the distribution of the estimators for different sizes of the spectrum and values of $\varphi$. A proper averaging of the power spectrum of the $\delta_n$ statistic needs to be performed for avoiding systematic errors in the estimation. Once the proper averaging is made the estimation of the fraction of observed levels has quite good accuracy for the two methods even for the lowest dimensions we consider $d=100$. However, the precision is generally better for the estimation using the power spectrum of the $\delta_n$ as compared to the estimation using the $\Delta_3$ statistic. This difference is clearly bigger for larger dimensions. Our results show that a careful analysis of the value of the fit in view of the ensemble distribution of the estimations is mandatory for understanding its actual significance and give a realistic error interval.