Manifestation of scale invariance in the spectral fluctuations of random matrices

We apply the Empirical Mode Decomposition method to extract intrinsic modes in the fluctuations of the spectra of random matrices. The hierarchy of the modes follows a power law with an exponent that changes smoothly in the whole range of the transition from the Gaussian orthogonal to the Gaussian diagonal ensemble. This property of scale invariance represents a new quantification of quantum chaos in systems that are characterized by the statistics of Random Matrix Theory.

Figure 1

Overall results of the logarithmic energy-timescale plots of the first 6 IMFs in the EMD-based analysis for GOE, GDE, and intermediate cases characterized by different values of $\sigma$ (see Table ). Each point corresponds with the ensemble average for the IMFs obtained for each value of $\sigma$, the corresponding error bars in both directions (energy and mean period) are represented with dashed lines. Different values of $\sigma$ corresponds with different lines. The fitted slopes ${\eta }_{\mathrm{emd}}$ are shown in the legend and plotted in dashed black lines. This result corresponds to the analysis of the ensemble with matrices whose dimension is 4000, a same type of behavior is obtained for the other dimensions.

Figure 2

Numerical results for the ensemble average of the ${\Delta }_3$ statistic obtained with a random-matrix model to describe the spectral statistic in intermediate situations between the limits GDE and GOE, respectively. The dashed black line and the black line are the limits while the $\sigma$ values are the intermediate cases. For each case, the error bars were calculated with the standard deviations. This result corresponds to the analysis of the ensemble with matrices whose dimension is 4000; a same type of behavior is obtained for 500 and 2000.