Eigenvalue Correlations Research Articles

Triple correlation and long gaps in the spectrum of flat tori

We evaluate the triple correlation of eigenvalues of the Laplacian on generic flat tori in an averaged sense. As two consequences we show that (a) the limit inferior (resp. limit superior) of the triple correlation is almost surely at most (resp. at least) Poissonian, and that (b) almost all flat tori contain infinitely many gaps in their spectrum that are at least 2.006 times longer than the average gap. The significance of the constant 2.006 lies in the fact that there exist sequences with Poissonian pair correlation and with consecutive spacings bounded uniformly from above by 2, as we also prove in this paper. Thus our result goes beyond what can be deduced solely from the Poissonian behavior of the pair correlation.

Meaningful local signalling in sinoatrial node identified by random matrix theory and PCA

The sinoatrial node (SAN) is the pacemaker of the heart. Recently calcium signals, believed to be crucially important in rhythm generation, have been imaged in intact SAN and shown to be heterogeneous in various regions of the SAN with a lot of analysis relying on visual inspection rather than mathematical tools. Here we apply methods of random matrix theory (RMT) developed for financial data and various biological data sets including β-cell collectives and electroencephalograms (EEG) to analyse correlations in SAN calcium signals using eigenvalues and eigenvectors of the correlation matrix. We use principal component analysis to locate signalling modules corresponding to localization properties the eigenvectors corresponding to high eigenvalues. We find that the top eigenvector captures the global behaviour of the SAN i.e. action potential (AP) induced calcium transient. In some cases, the eigenvector corresponding to the second highest eigenvalue yields a pacemaker region whose calcium signals predict the AP. Furthermore, using new analytic methods, we study the relationship between covariance coefficients and distance, and find that even inside the central zone, there are non-trivial long range correlations, indicating intercellular interactions in most cases. Lastly, we perform an analysis of nearest-neighbour eigenvalue distances and find that it coincides with universal Wigner surmise under all available experimental conditions, while the number variance, which captures eigenvalue correlations, is sensitive to experimental conditions. Thus RMT application to SAN allows to remove noise and the global effects of the AP-induced calcium transient and thereby isolate the local and meaningful correlations in calcium signalling.

FTIR fingerprinting combined with chemometrics of Guiera senegalensis leaf powder

Guiera senegalensis is a shrub that grows up to a height of 3 to 5 m and has tremendous medicinal properties. However, there is difficulty in the identification of medicinal plants leading to substitution of one plant for another. This problem can be addressed by establishing fingerprints of the medicinal plants. The aim of this research is to establish FTIR fingerprint of G. senegalensis leave. G. senegalensis leave were collected from the wild in Jigawa, Kaduna and Zamfara states, shade dried and size reduced. Samples (2 mg) were analysed in triplicate using Agilent technologies FTIR Carry 630 in the mid-IR region 4000-650 cm-1 at resolution 8 cm-1 with 16 scans. The level of similarity and differences was determined by Chemometrics using similarity and discriminant analysis. The level of differences between the samples was insignificant (p<0.05) as indicated by low values of eigenvalue (0.048) and canonical correlation (0.215) as well as high Wilks’ Lambda value (0.954). High level of similarity between the samples was further proven by their correlation and Tuckers’ congruence coefficients values, which were close to unity. However, significant (p<0.05) difference in the concentration of phytochemicals was observed between the samples. The Guiera senegalensis leaf samples were similar with respect to their phytochemical constituents however, differences in the phytochemical concentrations were observed.

Spectral properties of disordered interacting non-Hermitian systems

Non-hermitian systems have gained a lot of interest in recent years. However, notions of chaos and localization in such systems have not reached the same level of maturity as in the Hermitian systems. Here, we consider non-hermitian interacting disordered Hamiltonians and attempt to analyze their chaotic behavior or lack of it through the lens of the recently introduced non-hermitian analog of the spectral form factor and the complex spacing ratio. We consider three widely relevant non-hermitian models which are unique in their ways and serve as excellent platforms for such investigations. Two of the models considered are short-ranged and have different symmetries. The third model is long-ranged, whose hermitian counterpart has itself become a subject of growing interest. All these models exhibit a deep connection with the non-hermitian Random Matrix Theory of corresponding symmetry classes at relatively weak disorder. At relatively strong disorder, the models show the absence of complex eigenvalue correlation, thereby, corresponding to Poisson statistics. Our thorough analysis is expected to play a crucial role in understanding disordered open quantum systems in general.

Approximation formula for complex spacing ratios in the Ginibre ensemble.

Recently, Sá, Ribeiro, and Prosen [Phys. Rev. X 10, 021019 (2020)10.1103/PhysRevX.10.021019] introduced complex spacing ratios to analyze eigenvalue correlations in non-Hermitian systems. At present there are no analytical results for the probability distribution of these ratios in the limit of large system size. We derive an approximation formula for the Ginibre universality class of random matrix theory which converges exponentially fast to the limit of infinite matrix size. We also give results for moments of the distribution in this limit.

Universal microscopic spectrum of the unquenched QCD Dirac operator at finite temperature

In the ε-regime of chiral perturbation theory the spectral correlations of the Euclidean QCD Dirac operator close to the origin can be computed using random matrix theory. To incorporate the effect of temperature, a random matrix ensemble has been proposed, where a constant, deterministic matrix is added to the Dirac operator. Its eigenvalue correlation functions can be written as the determinant of a kernel that depends on temperature. Due to recent progress in this specific class of random matrix ensembles, featuring a deterministic, additive shift, we can determine the limiting kernel and correlation functions in this class, which is the class of polynomial ensembles. We prove the equivalence between this new determinantal representation of the microscopic eigenvalue correlation functions and existing results in terms of determinants of different sizes, for an arbitrary number of quark flavours, with and without temperature, and extend them to non-zero topology. These results all agree and are thus universal when measured in units of the temperature dependent chiral condensate, as long as we stay below the chiral phase transition.

Critical edge behavior in the singularly perturbed Pollaczek–Jacobi type unitary ensemble

We investigate the orthogonal polynomials associated with a singularly perturbed Pollaczek–Jacobi type weight [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Based on our observation, we find that this weight includes the symmetric constraint [Formula: see text]. Our main results obtained here include two aspects: (1) Strong asymptotics: we deduce strong asymptotics of monic orthogonal polynomials with respect to the above weight function in different regions in the complex plane when the polynomial degree [Formula: see text] goes to infinity. Because of the effect of [Formula: see text] for varying [Formula: see text], the asymptotic behavior in a neighborhood of point [Formula: see text] is described in terms of the Airy function as [Formula: see text], but the Bessel function as [Formula: see text]. Due to symmetry, the similar local asymptotic behavior near the singular point [Formula: see text] can be derived. (2) Limiting eigenvalue correlation kernels: We calculate the limit of the eigenvalue correlation kernel of the corresponding unitary random matrix ensemble in the bulk of the spectrum described by the sine kernel, and at both sides of hard edge, expressed as a Painlevé III kernel. Our analysis is based on the Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems.

Comprehensive Computational Investigation of the Barton-Kellogg Reaction for Both Alkyl and Aryl Systems.

The course of the Barton-Kellogg (BK) reaction for alkyl- and aryl-substituted substrates has been investigated at the DLPNO-CCSD(T)/def2-TZVPP//ωB97X-D/def2-TZVPP level of theory, with results compared to available experimental kinetic data. Through comparison with the unsubstituted parent system, the preference for the formation of 1,3,4-dihydrothiadiazole over the isomeric 1,2,3-dihydrothiadiazole was observed to result from reduced steric repulsion in the relevant transition-state structure. Nitrogen extrusion [retro-(3 + 2)-cycloaddition] from the intermediate dihydrothiadiazole was found to be the rate-determining step. The barrier for this process was, however, significantly lower for aromatic substrates, which is consistent with the difficulty in isolating aryl-substituted dihydrothiadiazoles. The electronic structure of the transient thiocarbonyl ylide was also investigated, highlighting the contradictory results from wave-function theory- and density functional theory-based methods. Correlation of unrestricted natural orbital eigenvalues with previous experimental models suggested that the dipole intermediates possess low diradical character and are therefore considered to be closed-shell species. Exergonic conrotatory electrocyclization of the dipole led to sterically congested thiirane products, even for very bulky systems (di-t-butyl). These results complement the recent work of Mlostoń et al. Finally, DLPNO-CCSD(T)//ωB97X-D was found to be a reliable method for estimating the feasibility of the BK reaction, which should assist experimentalists in the selection of viable substrates.

Signal Processing Techniques for Optical Transmission Based on Eigenvalue Communication

A minimum mean squared error (MMSE) equalizer is a way to effectively increase transmission performance for nonlinear Fourier transform (NFT) based communication systems. Other equalization schemes, based on nonlinear equalizer approaches or neural networks, are interesting for NFT transmission due to their ability to deal with nonlinear correlations of the NFTs' eigenvalues and their coefficients. We experimentally investigated single- and dual-polarization long haul transmission with several modulation schemes and compared different equalization techniques including joint detection equalization and the use of neural networks. We observed that joint detection equalization provides range increases for shorter transmission distances while having low numeric complexity. We could further achieve bit error rates (BER) under HD-FEC for significant longer transmission distances in comparison to no equalization with different equalizers.

Improving EIT-Based Visualizations of Two-Phase Flows Using an Eigenvalue Correlation Method

Gas–liquid two-phase flows are encountered in various industrial processes involving high temperatures and high pressures, which necessitates nonintrusive sensing for real-time imaging of phase distribution and flow parameters. In this context, this article presents an electrical impedance tomography (EIT)-based eigenvalue correlation method that allows extracting two-phase flow features, namely, the void fraction and the flow regime, which are used in turn to improve flow visualizations. Benefiting from the so-called full-scan excitation strategy, the eigenvalue correlation method has been devised in to estimate the phase fraction from EIT raw measurements. In this article, this method is refined and integrated into an image-enhancing procedure, which is illustrated and validated using dynamic experimental data. A total of 80 experiments are considered with water and air mass flow rates ranging from 1.58 to 79.43 kg/min and from 0.1 to 5.0 kg/min, respectively, covering slug, plug, stratified smooth, stratified wavy, and annular flows. Based on a preliminary system calibration and a raw image guess, the volume-averaged void fractions are then estimated using the proposed method and integrated into EIT-based images to form binarized tomograms relative to the acquisition time. The EIT tomograms, thus, obtained show an excellent agreement with some <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula> -ray reference measurements of the phase distribution.

Numerical Simulation of GUE Two-Point Correlation and Cluster Functions

Numerical simulations of the two-point eigenvalue correlation and cluster functions of the Gaussian unitary ensemble (GUE) are carried out directly from their definitions in terms of deltas functions. The simulations are compared with analytical results which follow from three analytical formulas for the two-point GUE cluster function: (i) Wigner's exact formula in terms of Hermite polynomials, (ii) Brezin and Zee's approximate formula which is valid for points with small enough separations and (iii) French, Mello and Pandey's approximate formula which is valid on average for points with large enough separations. It is found that the oscillations present in formulas (i) and (ii) are reproduced by the numerical simulations if the width of the function used to represent the delta function is small enough and that the non-oscillating behaviour of formula (iii) is approached as the width is increased.

Determinantal Structure and Bulk Universality of Conditional Overlaps in the Complex Ginibre Ensemble

In these proceedings we summarise how the determinantal structure for the conditional overlaps among left and right eigenvectors emerges in the complex Ginibre ensemble at finite matrix size. An emphasis is put on the underlying structure of orthogonal polynomials in the complex plane and its analogy to the determinantal structure of $k$-point complex eigenvalue correlation functions. The off-diagonal overlap is shown to follow from the diagonal overlap conditioned on $k\geq2$ complex eigenvalues. As a new result we present the local bulk scaling limit of the conditional overlaps away from the origin. It is shown to agree with the limit at the origin and is thus universal within this ensemble.

Eigenvalue repulsion and eigenvector localization in sparse non-Hermitian random matrices.

Complex networks with directed, local interactions are ubiquitous in nature and often occur with probabilistic connections due to both intrinsic stochasticity and disordered environments. Sparse non-Hermitian random matrices arise naturally in this context and are key to describing statistical properties of the nonequilibrium dynamics that emerges from interactions within the network structure. Here we study one-dimensional (1D) spatial structures and focus on sparse non-Hermitian random matrices in the spirit of tight-binding models in solid state physics. We first investigate two-point eigenvalue correlations in the complex plane for sparse non-Hermitian random matrices using methods developed for the statistical mechanics of inhomogeneous two-dimensional interacting particles. We find that eigenvalue repulsion in the complex plane directly correlates with eigenvector delocalization. In addition, for 1D chains and rings with both disordered nearest-neighbor connections and self-interactions, the self-interaction disorder tends to decorrelate eigenvalues and localize eigenvectors more than simple hopping disorder. However, remarkable resistance to eigenvector localization by disorder is provided by large cycles, such as those embodied in 1D periodic boundary conditions under strong directional bias. The directional bias also spatially separates the left and right eigenvectors, leading to interesting dynamics in excitation and response. These phenomena have important implications for asymmetric random networks and highlight a need for mathematical tools to describe and understand them analytically.

Evaluation of urban taxi-carpooling matching schemes based on entropy weight fuzzy matter-element

The evaluation of urban taxi-carpooling matching is the key index for the success of taxi carpooling. In this research, matter-element theory is introduced. Moreover, a matter-element evaluation model is established, and a matter-element evaluation algorithm based on fuzzy entropy weight is proposed. In accordance with the characteristics of the matching data, the algorithm takes the distance to the carpool destination, the distance to the carpool starting point, the carpool distance and the time to the carpool starting point in the passenger-matching scheme as the characteristic indexes of the evaluation category. A compound fuzzy matter-element matrix is established through an index fuzzy value method, and a matter-element matrix of a preferred membership degree and a fuzzy matter-element matrix of standard deviation are constructed. An index weight is determined through entropy weight method, and the closeness degree of the matter-element matrix is calculated. It avoids the shortcoming of traditional matter-element method which uses artificial determination of an eigenvalue index and correlation degree calculation. Experimental results show that the evaluation result of this algorithm is close to the actual evaluation result. In addition, the evaluation effect is better in this algorithm than in the traditional matter-element algorithm.

Preserving topology while breaking chirality: from chiral orthogonal to anti-symmetric Hermitian ensemble

We consider a parameter dependent ensemble of two real random matrices with Gaussian distribution. It describes the transition between the symmetry class of the chiral Gaussian orthogonal ensemble (Cartan class B|DI) and the ensemble of antisymmetric Hermitian random matrices (Cartan class B|D). It enjoys the special feature that, depending on the matrix dimension , it has exactly zero-mode for even (odd), throughout the symmetry transition. This ‘topological protection’ is reminiscent of properties of topological insulators. We show that our ensemble represents a Pfaffian point process which is typical for such transition ensembles. On a technical level, our results follow from the applicability of the Harish-Chandra integral over the orthogonal group. The matrix-valued kernel determining all eigenvalue correlation functions is explicitly constructed in terms of skew-orthogonal polynomials, depending on the topological index . These polynomials interpolate between Laguerre and even (odd) Hermite polynomials for , in terms of which the two limiting symmetry classes can be solved. Numerical simulations illustrate our analytical results for the spectral density and an expansion for the distribution of the smallest eigenvalue at finite .

Critical edge behavior in the perturbed Laguerre unitary ensemble and the Painlevé V transcendent

In this paper, we consider the perturbed Laguerre unitary ensemble described by the weight function of w ( x , t ) = ( x + t ) λ x α e − x with x ≥ 0 , t > 0 , α > 0 , α + λ + 1 > 0 . The Deift–Zhou nonlinear steepest descent approach is used to analyze the limit of the eigenvalue correlation kernel. It was found that under the double scaling s = 4 n t , n → ∞ , t → 0 such that s is positive and finite, at the hard edge, the limiting kernel can be described by the φ -function related to a third-order nonlinear differential equation, which is equivalent to a particular Painlevé V (shorted as P V ) transcendent via a simple transformation. Moreover, this P V transcendent is equivalent to a general Painlevé III transcendent. For large s , the P V kernel reduces to the Bessel kernel J α . For small s , the P V kernel reduces to another Bessel kernel J α + λ . At the soft edge, the limiting kernel is the Airy kernel as the classical Laguerre weight.

Full Dysonian dynamics of the complex Ginibre ensemble

We find stochastic equations governing eigenvalues and eigenvectors of a dynamical complex Ginibre ensemble reaffirming the intertwined role played between both sets of matrix degrees of freedom. We solve the accompanying Smoluchowski–Fokker–Planck equation valid for any initial matrix. We derive evolution equations for the averaged extended characteristic polynomial and for a class of k-point eigenvalue correlation functions. From the latter we obtain a novel formula for the eigenvector correlation function which we inspect for Ginibre and spiric initial conditions and obtain macro- and microscopic limiting laws.

Propagation of Singular Behavior for Gaussian Perturbations of Random Matrices

We study the asymptotic behavior of the eigenvalues of Gaussian perturbations of large Hermitian random matrices for which the limiting eigenvalue density vanishes at a singular interior point or vanishes faster than a square root at a singular edge point. First, we show that the singular behavior propagates macroscopically for sufficiently small Gaussian perturbations, and we describe the macroscopic eigenvalue behavior for Gaussian perturbations of critical size. Secondly, for sufficiently small Gaussian perturbations of unitary invariant random matrices, we prove that the microscopic eigenvalue correlations near the singular point are described by the same limiting kernel as in the unperturbed case. We also interpret our results in terms of nonintersecting Brownian paths with random starting positions, and we establish multi-time generalizations of the microscopic results.

Bulk universality for generalized Wigner matrices with few moments

In this paper we consider $$N \times N$$ real generalized Wigner matrices whose entries are only assumed to have finite $$(2 + \varepsilon )$$ th moment for some fixed, but arbitrarily small, $$\varepsilon > 0$$ . We show that the Stieltjes transforms $$m_N (z)$$ of these matrices satisfy a weak local semicircle law on the nearly smallest possible scale, when $$\eta = \mathfrak {I}(z)$$ is almost of order $$N^{-1}$$ . As a consequence, we establish bulk universality for local spectral statistics of these matrices at fixed energy levels, both in terms of eigenvalue gap distributions and correlation functions, meaning that these statistics converge to those of the Gaussian orthogonal ensemble in the large N limit.

Extremal eigenvalue correlations in the GUE minor process and a law of fractional logarithm

Let $\lambda^{(N)}$ be the largest eigenvalue of the $N\times N$ GUE matrix which is the $N$th element of the GUE minor process, rescaled to converge to the standard Tracy–Widom distribution. We consider the sequence $\{\lambda^{(N)}\}_{N\geq1}$ and prove a law of fractional logarithm for the $\limsup$: \[\limsup_{N\to\infty}\frac{\lambda^{({{N}})}}{(\log N)^{2/3}}=(\frac{1}{4})^{2/3}\qquad \mbox{almost surely}.\] For the $\liminf$, we prove the weaker result that there are constants $c_{1},c_{2}>0$ so that \[-c_{1}\leq\liminf_{N\to\infty}\frac{\lambda^{({{N}})}}{(\log N)^{1/3}}\leq-c_{2}\qquad \mbox{almost surely}.\] We conjecture that in fact, $c_{1}=c_{2}=4^{1/3}$.