Power-spectrum characterization of the continuous Gaussian ensemble
Physical Review E
The continuous Gaussian ensemble, also known as the -Gaussian or -Hermite ensemble, is a natural extension of the classical Gaussian ensembles of real ͑ =1͒, complex ͑ =2͒, or quaternion ͑ =4͒ matrices, where is allowed to take any positive value. From a physical point of view, this ensemble may be useful to describe transitions between different symmetries or to describe the terrace-width distributions of vicinal surfaces. Moreover, its simple form allows one to speed up and increase the
... increase the efficiency of numerical simulations dealing with large matrix dimensions. We analyze the long-range spectral correlations of this ensemble by means of the ␦ n statistic. We derive an analytical expression for the average power spectrum of this statistic, P k ␦ , based on approximated forms for the two-point cluster function and the spectral form factor. We find that the power spectrum of ␦ n evolves from P k ␦ ϰ 1 / k at =1 to P k ␦ ϰ 1 / k 2 at = 0. Relevantly, the transition is not homogeneous with a 1 / f ␣ noise at all scales, but heterogeneous with coexisting 1 / f and 1 / f 2 noises. There exists a critical frequency k c ϰ that separates both behaviors: below k c , P k ␦ follows a 1 / f power law, while beyond k c , it transits abruptly to a 1 / f 2 power law. For Ͼ 1 the 1 / f noise dominates through the whole frequency range, unveiling that the 1 / f correlation structure remains constant as we increase the level repulsion and reduce to zero the amplitude of the spectral fluctuations. All these results are confirmed by stringent numerical calculations involving matrices with dimensions up to 10 5 .