Random-matrix spectra as a time series

Abstract

Spectra of ordered eigenvalues of finite Random Matrices are interpreted as a time series. Dataadaptive techniques from signal analysis are applied to decompose the spectrum in clearly differentiated trend and fluctuation modes, avoiding possible artifacts introduced by standard unfolding techniques. The fluctuation modes are scale invariant and follow different power laws for Poisson and Gaussian ensembles, which already during the unfolding allows to distinguish the two cases.PACS numbers: 05.45. Tp,05.45.Mt,02.50.Sk The study of spectral fluctuations within the framework of Random Matrix Theory (RMT) is a standard tool in the statistical study of quantum chaos in the excitation spectra of quantum systems [1][2][3][4]. Recently, the approach has found new applications in many fields, such as in the study of eigenspectra of adjacency matrices of networks [5][6][7], and eigenspectra of empirical correlation matrices in finance [8][9][10], the climate [11], electro-and magnetoencephalography [12][13][14], and in complex systems [15]. The interest of the approach lies in the fact that the level density fluctuations ρ(E) = ρ(E) − ρ(E) around the smooth global density ρ(E) are universal and indicate the underlying symmetry class of the system [2,16]. On the other hand, the global level density ρ(E) is system dependent, and an unfolding procedure needs to be performed, to separate the global and the fluctuating parts [1]. The unfolding is straightforward if an analytical formula is known to describe the global level density ρ(E) for the system under study, such as e.g. the gaussian and semicircle distributions for Possion and GOE matrix ensembles from RMT [2], or the MarchenkoPastur distribution for the Laguerre ensemble of random Wishart correlation matrices [17]. However, such analytical formulae are formally only adequate in the asymptotic limit for spectra with an infinite number of levels. Often, an analytical form for ρ(E) is unknown, as is the case for adjacency matrices [5]. In practical cases, having finite, albeit large matrices, the usual approach is then to project the sequence of ordered eigenvalues into unfolded values E(n) → N [E(n)], using a smooth (often polynomial) approximation N (E) to the accumulated density , 5, 20]. After unfolding, the short-range and long-range correlations can be quantified using standard fluctuations measures such as the Nearest-Neighbour Spacing (NNS) distribution, number variance Σ 2 and ∆ 3 . In a recent approach, the unfolded fluctuations of the accumulated level density function N (E) = N (E) − N (E) (also called δ n function) *

Data-adaptive unfolding of nonergodic spectra: Two-Body Random Ensemble

“…. , r associated to the oscillating normal modes of the fluctuations E [21]. In the present calculation, we find one global mode (n T = 1) for the matrix ensembles in the ergodic regime, and two global modes (n T = 2) in the nonergodic regime.…”

“…In Ref. [21], we applied the above data-adaptive unfolding to ergodic Gaussian ensembles. On the one hand, the fluctuation part k = n T +1, .…”

Crossover in nonstandard random-matrix spectral fluctuations without unfolding

“…[10]. In order to obtain results for the traditional fluctuation measures, we perform the data-adaptive unfolding of the spectra E (m) (n) using the global part E(n) calculated when we applied SVD to the ensemble [17]. In Fig.…”

“…spectrum containing N = 2000 levels. The optimal ensemble size is M ≈ N/4, because for M ≫ N/4 there is a long tail of insignificant partial variances in the scree diagram, whereas for M ≪ N/4 the range of scales of the scree diagram becomes restrained, but the statistical properties do not depend on the particular choices of N and M[17]…”

Normal mode analysis of spectra of random networks

“…Besides, in order to calculate the standard spectral fluctuation measures from RMT, a data-adaptive unfolding can be realized. This unfolding method has the advantage of being a self-consisted method, defined in an intrinsic way, such that the global part of the spectra is obtained from the data itself, instead of being imposed extrinsically [1].…”