Nearest-neighbor Spacing Distribution Research Articles

Spacing ratio characterization of the spectra of directed random networks.

Previous literature on random matrix and network science has traditionally employed measures derived from nearest-neighbor level spacing distributions to characterize the eigenvalue statistics of random matrices. This approach, however, depends crucially on eigenvalue unfolding procedures, which in many situations represent a major hindrance due to constraints in the calculation, especially in the case of complex spectra. Here we study the spectra of directed networks using the recently introduced ratios between nearest and next-to-nearest eigenvalue spacing, thus circumventing the shortcomings imposed by spectral unfolding. Specifically, we characterize the eigenvalue statistics of directed Erdős-Rényi (ER) random networks by means of two adjacency matrix representations, namely, (1) weighted non-Hermitian random matrices and (2) a transformation on non-Hermitian adjacency matrices which produces weighted Hermitian matrices. For both representations, we find that the distribution of spacing ratios becomes universal for a fixed average degree, in accordance with undirected random networks. Furthermore, by calculating the average spacing ratio as a function of the average degree, we show that the spectral statistics of directed ER random networks undergoes a transition from Poisson to Ginibre statistics for model 1 and from Poisson to Gaussian unitary ensemble statistics for model 2. Eigenvector delocalization effects of directed networks are also discussed.

Geometrical and spectral study of β-skeleton graphs.

We perform an extensive numerical analysis of β-skeleton graphs, a particular type of proximity graphs. In a β-skeleton graph (BSG) two vertices are connected if a proximity rule, that depends of the parameter β∈(0,∞), is satisfied. Moreover, for β>1 there exist two different proximity rules, leading to lune-based and circle-based BSGs. First, by computing the average degree of large ensembles of BSGs we detect differences, which increase with the increase of β, between lune-based and circle-based BSGs. Then, within a random matrix theory (RMT) approach, we explore spectral and eigenvector properties of random BSGs by the use of the nearest-neighbor energy-level spacing distribution and the entropic eigenvector localization length, respectively. The RMT analysis allows us to conclude that a localization transition occurs at β=1.

Normal mode analysis of spectra of random networks

Several spectral fluctuation measures of random matrix theory (RMT) have been applied in the study of spectral properties of networks. However, the calculation of those statistics requires performing an unfolding procedure, which may not be an easy task. In this work, network spectra are interpreted as time series, and we show how their short and long-range correlations can be characterized without implementing any previous unfolding. In particular, we consider three different representations of Erdős–Rényi (ER) random networks: standard ER networks, ER networks with random-weighted self-edges, and fully random-weighted ER networks. In each case, we apply singular value decomposition (SVD) such that the spectra are decomposed in trend and fluctuation normal modes. We obtain that the fluctuation modes exhibit a clear crossover between the Poisson and the Gaussian orthogonal ensemble statistics when the average degree of ER networks changes. Moreover, by using the trend modes, we perform a data-adaptive unfolding to calculate, for comparison purposes, traditional fluctuation measures such as the nearest neighbor spacing distribution, number variance Σ2, as well as Δ3 and δn statistics. The thorough comparison of RMT short and long-range correlation measures make us identify the SVD method as a robust tool for characterizing random network spectra.

Eigenvalue statistics for generalized symmetric and Hermitian matrices

Random matrix theory predicts the level statistics of a Hamiltonian to exhibit either clustering or repulsion if the underlying dynamics is integrable or chaotic, respectively. In various physical systems it is also possible to observe intermediate spectral properties showing the transition between different symmetry classes. In this work, we study generalized random matrix ensembles by dropping the constraint of canonical invariance and considering different variances in the diagonal and off-diagonal elements. Tuning the relative value of the variances we show that the distributions of the level spacings exhibit intermediate level statistics. The nearest neighbour spacing (NNS) distributions can be computed for generalized symmetric matrices exhibiting crossover from clustering to GOE-like repulsion. The analysis is extended to matrices where the distributions of NNS as well as ratio of nearest neighbour spacing (RNNS) show similar crossovers. We show that it is possible to calculate NNS distributions for Hermitian matrices () where crossovers also take place between clustering and repulsion as in GUE. For large symmetric and Hermitian matrices we use interpolation between clustered and repulsive regimes to quantify the system size dependence of the crossover boundary.

Manifestation of crossover from RMT fluctuation to Poisson fluctuation in BSE sensex, a prototype of financial systems

We use the nearest neighbour spacing distribution (NNSD) of random matrix theory (RMT) as a tool to identify the type of spectral fluctuations of matrix ensemble constructed using high frequency BSE sensex values. NNSD shows Wigner distribution for ensemble formed with detrended index values. The detrended raw index values has the universal properties of the Gaussian orthogonal ensemble of random matrices. In contrast, NNSD shows Poisson distribution for ensemble formed using log transformed return of the index values which is corroborated by computation of the number variance. We have also shown the intermediate transitions connecting RMT and Poissonian fluctuations applying NNSD on the data sets generated using power transformation on the return index values. Log transformation is obtained in the asymptotic limit of the power transformation. This gradual changes in the spectral fluctuations from RMT to Poissonian type is reminiscent of localization theory results obtained in simulated experiments. On the contrary, the BSE Sensex values, approximately 2 million data for a period of 2006–2010, has been taken as a prototype of real systems representing an emerging economy of the world.

Connection of the nearest-neighbor spacing distribution and the local box-counting dimension for discrete sets.

In a recent work [Phys. Rev. E 97, 030202(R) (2018)10.1103/PhysRevE.97.030202], Sakhr and Nieminen (SN) solved a hypothesis formulated two decades ago, according to which the local box-counting dimension D_{box}(r) of a given energy spectrum, or more generally of a discrete set, should exclusively depend on the nearest-neighbor spacing distribution P(s) of the spectrum (set). SN found analytically this dependence, which led them to obtain closed formulas for the local box-counting dimension of Poisson spectra and of spectra belonging to Gaussian orthogonal, unitary, and symplectic ensembles. Here, first, we present a different derivation of the equation establishing the connection of D_{box}(r) and P(s) using the concept of surrogate spectrum. Although our equation is formally different to the SN result, we prove that both are equivalent. Second, we apply our equation to solve the inverse problem of determining the functional form of P(s) for spectra with real fractal structure and constant box-counting dimension D_{box}, and we find that P(s) should behave as a power-law of the spacing, with an exponent given by -(1+D_{box}). Finally, we present four applications or consequences of this last result: First, we provide a simple algorithm able to generate random fractal spectra with prescribed and constant D_{box}. Second, we calculate D_{box} for the sets given by the zeros of fractional Brownian motions, whose P(s) is known to have a power-law tail. Third, we also study D_{box}(r) for the zeros of fractional Gaussian noises, whose P(s) in known to present fat (but not power-law) tails, and that could be misinterpreted as real fractals. And finally, we present the calculation of D_{box} for the spectra of Fibonacci Hamiltonians, known to have fractal properties, simply by fitting their corresponding P(s) to a power-law without the need of applying a box-counting algorithm.

Universality classes of quantum chaotic dissipative systems

We study the ensemble of complex symmetric matrices. The ensemble is useful in the study of the effect of dissipation on systems with time-reversal invariance. We consider the nearest-neighbor spacing distribution and spacing ratio to investigate the fluctuation statistics and show that these statistics are similar to that of dissipative chaotic systems with time-reversal invariance. We show that, unlike cubic repulsion in eigenvalues of Ginibre matrices, this ensemble exhibits a weaker repulsion. The nearest-neighbor spacing distribution exhibits for small spacings. We verify our results for quantum kicked rotor with time-reversal invariance. We show that the rotor exhibits similar spacing distribution in dissipative regime. We also discuss a random matrix model for transition from the time-reversal invariant to the broken case.

Experimental investigation of the elastic enhancement factor in a microwave cavity emulating a chaotic scattering system with varying openness.

A characteristic of chaotic scattering is the excess of elastic over inelastic scattering processes quantified by the elastic enhancement factor F_{M}(T,γ), which depends on the number of open channels M, the average transmission coefficient T, and internal absorption γ. Using a microwave cavity with the shape of a chaotic quarter-bow-tie billiard, we study the elastic enhancement factor experimentally as a function of the openness, which is defined as the ratio of the Heisenberg time and the Weisskopf (dwell) time and is directly related to M and the size of internal absorption. In the experiments 2≤M≤9 open channels with an average transmission coefficient 0.34<T<0.98 and moderate internal absorption strength in the range of γ=0.9-2.8 are achieved. The experimental results for the enhancement factor are shown to agree well with random matrix theory predictions. Furthermore, in order to corroborate the wave-chaotic features of the microwave system, the spectral fluctuation properties are studied for M=2 channels. Agreement with those exhibited by typical, fully chaotic systems is illustrated, which is exemplary for the nearest-neighbor spacing distribution and the average power spectrum. Here we take into account the incompleteness of the sequence of resonance frequencies ascribed to the small yet nonvanishing internal absorption.

Two-neutron transfer reactions and the quantum chaos measure of nuclear spectra

A new statistical interpretation of the nuclear collective states is suggested and applied recently in rare earths and actinide nuclei by the two-neutron transfer reactions in terms of the nearest neighbor-spacing distributions (NNSDs). Experimental NNSDs were obtained by using the complete and pure sequences of the collective states through an unfolding procedure. The two-neutron transfer reactions allow to obtain such a sequence of the collective states that meets the requirements for a statistical analysis. Their theoretical analysis is based on the linear approximation of a repulsion level density within the Wigner-Dyson theory. This approximation is successful to evaluate separately the Wigner chaos and Poisson order contributions. We found an intermediate behavior of NNSDs between the Wigner and Poisson limits. NNSDs turn out to be shifted from a chaos to order with increasing the length of spectra and the angular momentum of collective states. Perspectives for the statistical analysis of the symmetry breaking of states with the fixed projection of angular momenta $K$ are discussed.

Random-matrix behavior of quantum nonintegrable many-body systems with Dyson's three symmetries.

We propose a one-dimensional nonintegrable spin model with local interactions that covers Dyson's three symmetry classes (classes A, AI, and AII) depending on the values of parameters. We show that the nearest-neighbor spacing distribution in each of these classes agrees with that of random matrices having the same symmetry. By investigating the ratios between the standard deviations of diagonal and off-diagonal matrix elements, we numerically find that they are universal, depending only on symmetries of the Hamiltonian and an observable, as predicted by random matrix theory. These universal ratios are evaluated from long-time dynamics of small isolated quantum systems.

Simple approach to the chaos-order contributions and symmetry breaking in nuclear spectra

The simple one-parameter nearest neighbor-spacing distribution (NNSD) is suggested for statistical analysis of nuclear spectra. This distribution is derived within the Wigner-Dyson approach in the linear approximation for the level repulsion density of quantum states. The obtained NNSD gives the individual information on the Wigner and Poisson contributions in agreement with that of the statistical experimental distributions of collective states in deformed nuclei. Using this NNSD, one finds that the symmetry breaking due to the fixing of projections of the angular momentum of collective states enhances a chaos as a shift of the NNSD from the Poisson to Wigner distribution behavior.

Damped transient response of the giant dipole resonance in the lead region

We analyze thoroughly the statistical properties of the 1- spectrum in energy interval 9.5-18.5 MeV for Hg206, Pb204,206,208, and Po210 nuclei. To this aim we use the Skyrme interaction SLy4 and the volume pairing interaction, treated in the BCS approximation, as our model Hamiltonian to create a single-particle spectrum and to analyze excited states. The excited states are obtained in the quasiparticle random phase approximation taking into account the coupling between one- and two-phonon states. The obtained spectra (studied by means of the nearest-neighbour spacing distribution, Δ3-statistics and the Porter-Thomas distribution) indicate on the onset of statistical properties close to those of the Gaussian Orthogonal Ensembles. The comparison of the results, obtained with the aid of the coupling calculated microscopically and by means of the Gaussian random distribution, demonstrates a close similarity in the description of the spreading widths of the Isovector Dipole Giant Resonanse of the considered nuclei. Furthermore, we show that employing the random distribution for the coupling between microscopic one-phonon states and two-phonon states, generated by the Gaussian Orthogonal Ensembles distribution, a good agreement is also obtained with the microscopic description of the decay widths. (Less)

Statistical analysis of excitation energies in actinide and rare-earth nuclei

Statistical analysis of distributions of the collective states in actinide and rare-earth nuclei is performed in terms of the nearest-neighbor spacing distribution (NNSD). Several approximations, such as the linear approach to the level repulsion density and that suggested by Brody to the NNSDs were applied for the analysis. We found an intermediate character of the experimental spectra between the order and the chaos for a number of rare-earth and actinide nuclei. The spectra are closer to the Wigner distribution for energies limited by 3 MeV, and to the Poisson distribution for data including higher excitation energies and higher spins. The latter result is in agreement with the theoretical calculations. These features are confirmed by the cumulative distributions, where the Wigner contribution dominates at smaller spacings while the Poisson one is more important at larger spacings, and our linear approach improves the comparison with experimental data at all desired spacings.

Local box-counting dimensions of discrete quantum eigenvalue spectra: Analytical connection to quantum spectral statistics.

Two decades ago, Wang and Ong, [Phys. Rev. A 55, 1522 (1997)]10.1103/PhysRevA.55.1522 hypothesized that the local box-counting dimension of a discrete quantum spectrum should depend exclusively on the nearest-neighbor spacing distribution (NNSD) of the spectrum. In this Rapid Communication, we validate their hypothesis by deriving an explicit formula for the local box-counting dimension of a countably-infinite discrete quantum spectrum. This formula expresses the local box-counting dimension of a spectrum in terms of single and double integrals of the NNSD of the spectrum. As applications, we derive an analytical formula for Poisson spectra and closed-form approximations to the local box-counting dimension for spectra having Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE), and Gaussian symplectic ensemble (GSE) spacing statistics. In the Poisson and GOE cases, we compare our theoretical formulas with the published numerical data of Wang and Ong and observe excellent agreement between their data and our theory. We also study numerically the local box-counting dimensions of the Riemann zeta function zeros and the alternate levels of GOE spectra, which are often used as numerical models of spectra possessing GUE and GSE spacing statistics, respectively. In each case, the corresponding theoretical formula is found to accurately describe the numerically computed local box-counting dimension.

Experimental evidence of chaos in the bound states of 208Pb

Theoretical calculations, especially shell-model calculations, have shown a strongly chaotic behavior of bound states at higher excitation energy, in regions of high level density. However, it had not been possible up to now to observe chaos in the experimental bound energy levels of any single nucleus. In this paper we study the spectral fluctuations of the 208Pb nucleus using the complete experimental spectrum of 151 states up to excitation energies of 6.20 MeV. For natural parity states the results are very close to the predictions of Random Matrix Theory (RMT) for the nearest-neighbor spacing distribution. By contrast, the results for unnatural parity states are far from RMT behavior. We interpret these results as a consequence of the strength of the residual interaction in 208Pb, which, according to experimental data, is much stronger for natural than for unnatural parity states. In addition our results show that chaotic and non-chaotic nuclear states coexist in the same energy region of the spectrum.

Weighted random-geometric and random-rectangular graphs: spectral and eigenfunction properties of the adjacency matrix

Within a random-matrix-theory approach, we use the nearest-neighbor energy level spacing distribution $P(s)$ and the entropic eigenfunction localization length $\ell$ to study spectral and eigenfunction properties (of adjacency matrices) of weighted random--geometric and random--rectangular graphs. A random--geometric graph (RGG) considers a set of vertices uniformly and independently distributed on the unit square, while for a random--rectangular graph (RRG) the embedding geometry is a rectangle. The RRG model depends on three parameters: The rectangle side lengths $a$ and $1/a$, the connection radius $r$, and the number of vertices $N$. We then study in detail the case $a=1$ which corresponds to weighted RGGs and explore weighted RRGs characterized by $a\sim 1$, i.e.~two-dimensional geometries, but also approach the limit of quasi-one-dimensional wires when $a\gg1$. In general we look for the scaling properties of $P(s)$ and $\ell$ as a function of $a$, $r$ and $N$. We find that the ratio $r/N^\gamma$, with $\gamma(a)\approx -1/2$, fixes the properties of both RGGs and RRGs. Moreover, when $a\ge 10$ we show that spectral and eigenfunction properties of weighted RRGs are universal for the fixed ratio $r/{\cal C}N^\gamma$, with ${\cal C}\approx a$.

Manifestations of isospin in nearest neighbor spacing distributions for the f-p model space

The strong interactions are charge independent. If we limit ourselves to the strong interactions, we have the isospin [Formula: see text] as a good quantum number. Here, we consider the lack of level repulsion of states of different isospin and how this effect manifests in nearest neighbor spacing (NNS) histograms, which provide a visual and statistical context in which to study distributions of energy level spacings. In particular, we study nucleons in the [Formula: see text] model space for the nucleus [Formula: see text]. We also compare NNS distributions produced in the isospin formalism to distributions produced in the proton–neutron formalism.

Statistical behavior of time dynamics evolution of HIV infection

We use the tools of the random matrix theory (RMT) to investigate the statistical behavior of the evolution of human immunodeficiency virus (HIV) infection. By means of the nearest-neighbor spacing distribution we have identified four distinct regimes of the evolution of HIV infection. We verified that at the beginning of the so-called clinical latency phase the concentration of infected cells grows slowly and evolves in a correlated way. This regime is followed by another one in which the correlation is lost and that in turn leads the system to a regime in which the increase of infected cells is faster and correlated. In the final phase, the one in which acquired immunodeficiency syndrome (AIDS) is stablished, the system presents maximum correlation as demonstrated by GOE distribution.

GOE-GUE-Poisson transitions in the nearest-neighbor spacing distribution of magnetoexcitons.

Recent investigations on the Hamiltonian of excitons by Schweiner et al.. [Phys. Rev. Lett. 118, 046401 (2017)]PRLTAO0031-900710.1103/PhysRevLett.118.046401 revealed that the combined presence of a cubic band structure and external fields breaks all antiunitary symmetries. The nearest-neighbor spacing distribution of magnetoexcitons can exhibit Poissonian statistics, the statistics of a Gaussian orthogonal ensemble (GOE), or a Gaussian unitary ensemble (GUE) depending on the system parameters. Hence, magnetoexcitons are an ideal system to investigate the transitions between these statistics. Here we investigate the transitions between GOE and GUE statistics and between Poissonian and GUE statistics by changing the angle of the magnetic field with respect to the crystal lattice and by changing the scaled energy known from the hydrogen atom in external fields. Comparing our results with analytical formulas for these transitions derived with random matrix theory, we obtain a very good agreement and thus confirm the Wigner surmise for the exciton system.

Excited-state quantum phase transitions in a two-fluid Lipkin model

Background: Composed systems have became of great interest in the framework of the ground state quantum phase transitions (QPTs) and many of their properties have been studied in detail. However, in these systems the study of the so called excited-state quantum phase transitions (ESQPTs) have not received so much attention. Purpose: A quantum analysis of the ESQPTs in the two-fluid Lipkin model is presented in this work. The study is performed through the Hamiltonian diagonalization for selected values of the control parameters in order to cover the most interesting regions of the system phase diagram. [Method:] A Hamiltonian that resembles the consistent-Q Hamiltonian of the interacting boson model (IBM) is diagonalized for selected values of the parameters and properties such as the density of states, the Peres lattices, the nearest-neighbor spacing distribution, and the participation ratio are analyzed. Results: An overview of the spectrum of the two-fluid Lipkin model for selected positions in the phase diagram has been obtained. The location of the excited-state quantum phase transition can be easily singled out with the Peres lattice, with the nearest-neighbor spacing distribution, with Poincar\'e sections or with the participation ratio. Conclusions: This study completes the analysis of QPTs for the two-fluid Lipkin model, extending the previous study to excited states. The ESQPT signatures in composed systems behave in the same way as in single ones, although the evidences of their presence can be sometimes blurred. The Peres lattice turns out to be a convenient tool to look into the position of the ESQPT and to define the concept of phase in the excited states realm.