Randomly incomplete spectra and intermediate statistics

Disordered beta thinned ensemble with applications

It recently has been found that methods of the statistical theories of spectra can be a useful tool in the analysis of spectra far from levels of Hamiltonian systems. The purpose of the present study is to deepen this kind of approach by performing a more comprehensive spectral analysis that measures both the local- and long-range statistics. We have found that, as a common feature, spectra of this kind can exhibit a situation in which local statistics are relatively quenched while the long-range ones show large fluctuations. By combining three extensions of the standard random matrix theory (RMT) and considering long spectra, we demonstrate that this phenomenon occurs when disorder and level incompleteness are introduced in an RMT spectrum. Consequently, the long-range statistics follow Taylor's law, suggesting the presence of a fluctuation scaling (FS) mechanism in this kind of spectra. Applications of the combined ensemble are then presented for spectra originate from several very diverse areas, including complex networks, COVID-19 time series, and quantitative linguistics, which demonstrate that short- and long-range statistics reflect the rigid and elastic characteristics of a given spectrum, respectively. These observations may shed some light on spectral data classification.

Large gap asymptotics for the generating function of the sine point process

We consider the generating function of the sine point process on m consecutive intervals. It can be written as a Fredholm determinant with discontinuities, or equivalently as the convergent series ∑ k 1 , … , k m ⩾ 0 P ∩ j = 1 m # { points in the jth interval } = k j ∏ j = 1 m s j k j , where s 1 , … , s m ∈ [ 0 , + ∞ ) . In particular, we can deduce from it joint probabilities of the counting function of the process. In this work, we obtain large gap asymptotics for the generating function, which are asymptotics as the size of the intervals grows. Our results are valid for an arbitrary integer m , in the cases where all the parameters s 1 , … , s m , except possibly one, are positive. This generalizes two known results: (1) a result of Basor and Widom, which corresponds to m = 1 and s 1 > 0 , and (2) the case m = 1 and s 1 = 0 for which many authors have contributed. We also present some applications in the context of thinning and conditioning of the sine process.

Missing-level statistics in a dissipative microwave resonator with partially violated time-reversal invariance

We report on the experimental investigation of the fluctuation properties in the resonance frequency spectra of a flat resonator simulating a dissipative quantum billiard subject to partial time-reversal-invariance violation (TIV) which is induced by two magnetized ferrites. The cavity has the shape of a quarter bowtie billiard of which the corresponding classical dynamics is chaotic. Due to dissipation it is impossible to identify a complete list of resonance frequencies. Based on a random-matrix theory approach we derive analytical expressions for statistical measures of short- and long-range correlations in such incomplete spectra interpolating between the cases of preserved time-reversal invariance and complete TIV and demonstrate their applicability to the experimental spectra.

Missing-level statistics in classically chaotic quantum systems with symplectic symmetry

We present experimental and theoretical results for the fluctuation properties in the incomplete spectra of quantum systems with symplectic symmetry and a chaotic dynamics in the classical limit. To obtain theoretical predictions, we extend the random-matrix theory (RMT) approach introduced in Bohigas and Pato [O. Bohigas and M. P. Pato, Phys. Rev. E 74, 036212 (2006)PLEEE81539-375510.1103/PhysRevE.74.036212] for incomplete spectra of quantum systems with orthogonal symmetry. We validate these RMT predictions by randomly extracting a fraction of levels from complete sequences obtained numerically for quantum graphs and experimentally for microwave networks with symplectic symmetry and then apply them to incomplete experimental spectra to demonstrate their applicability. Independently of their symmetry class, quantum graphs exhibit nongeneric features which originate from nonuniversal contributions. Part of the associated eigenfrequencies can be identified in the level dynamics of parameter-dependent quantum graphs and extracted, thereby yielding spectra with systematically missing eigenfrequencies. We demonstrate that, even though the RMT approach relies on the assumption that levels are missing at random, it is possible to determine the fraction of missing levels and assign the appropriate symmetry class by comparison of their fluctuation properties with the RMT predictions.

Sensitivity of the spectral form factor to short-range level statistics

The spectral form factor is a dynamical probe for level statistics of quantum systems. The early-time behavior is commonly interpreted as a characterization of two-point correlations at large separation. We argue that this interpretation can be too restrictive by indicating that the self-correlation imposes a constraint on the spectral form factor integrated over time. More generally, we indicate that each expansion coefficient of the two-point correlation function imposes a constraint on the properly weighted time-integrated spectral form factor. We discuss how these constraints can affect the interpretation of the spectral form factor as a probe for ergodicity. We propose a probe, which eliminates the effect of the constraint imposed by the self-correlation. The use of this probe is demonstrated for a model of randomly incomplete spectra and a Floquet model supporting many-body localization.

Long-range correlations in rectangular cavities containing pointlike perturbations

We investigated experimentally the short- and long-range correlations in the fluctuations of the resonance frequencies of flat, rectangular microwave cavities that contained antennas acting as pointlike perturbations. We demonstrate that their spectral properties exhibit the features typical for singular statistics. Hitherto, only the nearest-neighbor spacing distribution had been studied. In addition we considered statistical measures for the long-range correlations and analyzed power spectra. Thereby we could corroborate that the spectral properties change to semi-Poisson statistic with increasing microwave frequency. Furthermore, the experimental results are shown to be well described by a model applicable to billiards containing a zero-range perturbation [T. Tudorovskiy et al., New J. Phys. 12, 123021 (2010)NJOPFM1367-263010.1088/1367-2630/12/12/123021].

Spectral statistics of molecular resonances in erbium isotopes: How chaotic are they?

We perform a comprehensive analysis of the spectral statistics of the molecular resonances in (166)Er and (168)Er observed in recent ultracold collision experiments [Frisch et al., Nature (London) 507, 475 (2014)] with the aim of determining the chaoticity of this system. We calculate different independent statistical properties to check their degree of agreement with random matrix theory (RMT), and analyze if they are consistent with the possibility of having missing resonances. The analysis of the short-range fluctuations as a function of the magnetic field points to a steady increase of chaoticity until B∼30 G. The repulsion parameter decreases for higher magnetic fields, an effect that can be interpreted as due to missing resonances. The analysis of long-range fluctuations allows us to be more quantitative and estimate a 20%-25% fraction of missing levels. Finally, a study of the distribution of resonance widths provides additional evidence supporting missing resonances of small width compared with the experimental magnetic field resolution. We conclude that further measurements with increased resolution will be necessary to give a final answer to the problem of missing resonances and the agreement with RMT.

Hyperbolic disordered ensembles of random matrices

Using the recently introduced simple procedure of dividing Gaussian matrices by a positive random variable, a family of random matrices is generated characterized by a behavior ruled by the generalized hyperbolic distribution. The spectral density evolves from the semicircle law to a Gaussian-like behavior while concomitantly, the local fluctuations show a transition from the Wigner-Dyson to the Poisson statistics. Long range statistics such as number variance exhibit large fluctuations typical of nonergodic ensembles.

Magnetic field control of the quantum chaotic dynamics of hydrogen analogs in an anisotropic crystal field

We report magnetic field control of the quantum chaotic dynamics of hydrogen analogues in an anisotropic solid state environment. The chaoticity of the system dynamics was quantified by means of energy level statistics. We analyzed the magnetic field dependence of the statistical distribution of the impurity energy levels and found a smooth transition between the Poisson limit and the Wigner limit, i.e., transition between regular Poisson and fully chaotic Wigner dynamics. The effect of the crystal field anisotropy on the quantum chaotic dynamics, which manifests itself in characteristic transitions between regularity and chaos for different field orientations, was demonstrated.

Deformations of the Tracy-Widom distribution

In random matrix theory, the Tracy-Widom (TW) distribution describes the behavior of the largest eigenvalue. We consider here two models in which TW undergoes transformations. In the first one disorder is introduced in the Gaussian ensembles by superimposing an external source of randomness. A competition between TW and a normal (Gaussian) distribution results, depending on the spreading of the disorder. The second model consists of removing at random a fraction of (correlated) eigenvalues of a random matrix. The usual formalism of Fredholm determinants extends naturally. A continuous transition from TW to the Weilbull distribution, characteristic of extreme values of an uncorrelated sequence, is obtained.

Missing levels in acoustic resonators

It is shown that the deviations of the experimental statistics of six chaotic acoustic resonators from Wigner-Dyson random matrix theory predictions are explained by a recent model of random missing levels. In these resonatorsa made of aluminum plates a the larger deviations occur in the spectral rigidity (SRs) while the nearest-neighbor distributions (NNDs) are still close to the Wigner surmise. Good fits to the experimental NNDs and SRs are obtained by adjusting only one parameter, which is the fraction of remaining levels of the complete spectra. For two Sinai stadiums, one Sinai stadium without planar symmetry, two triangles, and a sixth of the three-leaf clover shapes, was found that 7%, 4%, 7%, and 2%, respectively, of eigenfrequencies were not detected.

Spectral-fluctuations test of the quark-model baryon spectrum

We study the low-lying baryon spectrum (up to 2.2 GeV) provided by experiments and different quark models using statistical tools which allow us to postulate the existence of missing levels in spectra. We confirm that the experimental spectrum is compatible with random matrix theory, the paradigmatic model of quantum chaos, and we find that the quark models are more similar to a Poisson distribution, which is not compatible with what should be expected in a correlated spectrum. From our analysis it stems that the spectral fluctuation properties of quark-model spectra are incompatible with experimental data. This result can be used to enlighten the problem of missing resonances.