Wigner's Law Research Articles

Local semicircle law and Gaussian fluctuation for Hermite <italic>β</italic> ensemble

Let β > 0 and consider an n -point process λ1, λ2,..., λn from Hermite β ensemble on the real line R. Dumitriu and Edelman discovered a tri-diagonal matrix model and established the global Wigner semicircle law for normalized empirical measures. In this paper, we prove that the average number of states in a small interval in the bulk converges in probability when the length of the interval is larger than √log n , i.e., local semicircle law holds. And the number of positive states in (0,∞) is proved to uctuate normally around its mean n /2 with variance like log n / π 2 β . The proofs rely largely on the way invented by Valko and Virag of counting states in any interval and the classical martingale argument.

Randomly weighted averages with beta random proportions

A weighted average of two independent continuous random variables X1 and X2 with random proportions obtained by Beta distribution is introduced. A formula between the Stieltjes transforms of the distribution functions of the weighted averages and X1, X2 is established. We show that, among some other distributions, the Cauchy distribution and the power semicircle distribution can be characterized in a particular way by means of this construction.

On distribution of randomly ordered uniform incremental weighted averages: Divided difference approach

In this article we employ certain techniques in divided differences to relate the generalized Stieltjes transform of the distribution of a randomly weighted average of independent random variables X1,…,Xm to the generalized Stieltjes transforms of the distribution functions F1,…,Fm; Xi∼Fi,i=1,…,m. The random weights are assumed to be cuts of [0,1] by m−1 ordered statistics of independent and identically uniformly distributed random variables U1,…,Un on [0,1]; m≤n. Soltani and Homei (2009) treated the case m=n using the Schwartz distribution theory. We identified fairly large classes of randomly weighted average distributions by their generalized Stieltjes transforms; in particular including the uniform, Wigner and certain power semicircle distributions.

Rigidity of eigenvalues of generalized Wigner matrices

Consider N × N Hermitian or symmetric random matrices H with independent entries, where the distribution of the ( i , j ) matrix element is given by the probability measure ν i j with zero expectation and with variance σ i j 2 . We assume that the variances satisfy the normalization condition ∑ i σ i j 2 = 1 for all j and that there is a positive constant c such that c ⩽ N σ i j 2 ⩽ c − 1 . We further assume that the probability distributions ν i j have a uniform subexponential decay. We prove that the Stieltjes transform of the empirical eigenvalue distribution of H is given by the Wigner semicircle law uniformly up to the edges of the spectrum with an error of order ( N η ) − 1 where η is the imaginary part of the spectral parameter in the Stieltjes transform. There are three corollaries to this strong local semicircle law: (1) Rigidity of eigenvalues: If γ j = γ j , N denotes the classical location of the j-th eigenvalue under the semicircle law ordered in increasing order, then the j-th eigenvalue λ j is close to γ j in the sense that for some positive constants C, c P ( ∃ j : | λ j − γ j | ⩾ ( log N ) C log log N [ min ( j , N − j + 1 ) ] − 1 / 3 N − 2 / 3 ) ⩽ C exp [ − ( log N ) c log log N ] for N large enough. (2) The proof of Dysonʼs conjecture (Dyson, 1962 [15]) which states that the time scale of the Dyson Brownian motion to reach local equilibrium is of order N − 1 up to logarithmic corrections. (3) The edge universality holds in the sense that the probability distributions of the largest (and the smallest) eigenvalues of two generalized Wigner ensembles are the same in the large N limit provided that the second moments of the two ensembles are identical.

Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices

We study the fluctuations of the matrix entries of regular functions of Wigner random matrices in the limit when the matrix size goes to infinity. In the case of the Gaussian ensembles (GOE and GUE) this problem was considered by A.Lytova and L.Pastur in J. Stat. Phys., v.134, 147-159 (2009). Our results are valid provided the off-diagonal matrix entries have finite fourth moment, the diagonal matrix entries have finite second moment, and the test functions have four continuous derivatives in a neighborhood of the support of the Wigner semicircle law.

Statistics of Group Delays in Multimode Fiber With Strong Mode Coupling

The modal group delays (GDs) are a key property governing the dispersion of signals propagating in a multimode fiber (MMF). A MMF is in the strong-coupling regime when the total length of the MMF is much greater than the correlation length over which local principal modes can be considered constant. In this regime, the GDs can be described as the eigenvalues of zero-trace Gaussian unitary ensemble, and the probability density function (p.d.f.) of the GDs is the eigenvalue distribution of the ensemble. For fibers with two to seven modes, the marginal p.d.f. of the GDs is derived analytically. For fibers with a large number of modes, this p.d.f. is shown to approach a semicircle distribution. In the strong-coupling regime, the delay spread is proportional to the square root of the number of independent sections, or the square root of the overall fiber length. This revision also made clarification to the original paper, the group delay statistics is also extended to high number of modes.

A note on eigenvalues of random block Toeplitz matrices with slowly growing bandwidth

This paper can be thought of as a remark of Li et al. (2010), where the authors studied the eigenvalue distribution μ X N of random block Toeplitz band matrices with given block order m . In this paper, we will give explicit density functions of lim N → ∞ μ X N when the bandwidth grows slowly. In fact, these densities are exactly the normalized one-point correlation functions of m × m Gaussian unitary ensemble (GUE for short). The series { lim N → ∞ μ X N ∣ m ∈ N } can be seen as a transition from the standard normal distribution to semicircle distribution. We also show a similar relationship between GOE and block Toeplitz band matrices with symmetric blocks.

Product of Ginibre matrices: Fuss-Catalan and Raney distributions

Squared singular values of a product of s square random Ginibre matrices are asymptotically characterized by probability distributions P(s)(x), such that their moments are equal to the Fuss-Catalan numbers of order s. We find a representation of the Fuss-Catalan distributions P(s)(x) in terms of a combination of s hypergeometric functions of the type (s)F(s-1). The explicit formula derived here is exact for an arbitrary positive integer s, and for s=1 it reduces to the Marchenko-Pastur distribution. Using similar techniques, involving the Mellin transform and the Meijer G function, we find exact expressions for the Raney probability distributions, the moments of which are given by a two-parameter generalization of the Fuss-Catalan numbers. These distributions can also be considered as a two-parameter generalization of the Wigner semicircle law.

Level density and level-spacing distributions of random, self-adjoint, non-Hermitian matrices

We investigate the level density σ(x) and the level-spacing distribution p(s) of random matrices M = AF ≠ M{†}, where F is a (diagonal) inner product and A is a random, real, symmetric or complex, Hermitian matrix with independent entries drawn from a probability distribution q(x) with zero mean and finite higher moments. Although not Hermitian, the matrix M is self-adjoint with respect to F and thus has purely real eigenvalues. We find that the level density σ{F}(x) is independent of the underlying distribution q(x) and solely characterized by F, and therefore generalizes the Wigner semicircle distribution σ{W}(x). We find that the level-spacing distributions p(s) are independent of q(x), and are dependent upon both the inner product F and whether A is real or complex, and therefore generalize the Wigner surmise for level spacing. Our results suggest F-dependent generalizations of the well-known Gaussian Orthogonal Ensemble and Gaussian Unitary Ensemble classes.

Large Deviations for Random Spectral Measures and Sum Rules

We prove a Large Deviation Principle for the random spec- tral measure associated to the pair $(H_N; e)$ where $H_N$ is sampled in the GUE(N) and e is a fixed unit vector (and more generally in the $\beta$- extension of this model). The rate function consists of two parts. The contribution of the absolutely continuous part of the measure is the reversed Kullback information with respect to the semicircle distribution and the contribution of the singular part is connected to the rate function of the extreme eigenvalue in the GUE. This method is also applied to the Laguerre and Jacobi ensembles, but in thoses cases the expression of the rate function is not so explicit.

Higher-dimensional conformal field theories in the Coulomb branch

We use the AdS/CFT correspondence to study flows of N=4 SYM to non-conformal theories. The dual geometries can be seen as sourced by a Wigner's semicircle distribution of D3 branes. We consider two cases, the first case corresponds to a point in the Coulomb branch and the theory flows to a six-dimensional conformal field theory. In the second case a mass is introduced for a hypermultiplet and the theory flows to a five-dimensional conformal field theory. We argue from the gravity and the field theory side that the low energy theories correspond to the (2,0) theory in six dimensions and to a theory with exceptional global symmetry E1 in five dimensions.

Rovibrational quenching of BH in ultracold 3He collisions

The interaction potential of a He–BH complex is investigated by the coupled-cluster single-double plus perturbative triples (CCSD (T)) method and an augmented correlation consistent polarized valence (aug-cc-pV)5Z basis set extended with a set of (3s3p2d1f1g) midbond functions. Using the five two-dimensional model potentials, the first three-dimensional interaction potential energy surface is constructed by interpolating along (r–re) by using a fourth-order polynomial. The cross sections for the rovibrational relaxation of BH in cold and ultracold collisions with 3He atom are calculated based on the three-dimensional potential. The results show that the Δν = −1 transition is more efficient than the Δν = −2 transition, and that the process of relaxation takes place mainly between rotational energy levels with the same vibration state and the Δj = −1 transition is the most efficient. The zero temperature quenching rate coefficient is finite as predicted by Wigner's law. The resonance is found to take place around 0.1–1 cm−1 translational energy, which gives rise to a step in the rate coefficients for temperatures around 0.1–1 K. The final rotational distributions in the state ν = 0 resulting from the quenching of state (ν = 1, j = 0) at three energies corresponding to the three different regimes are also given.

On the non-classical infinite divisibility of power semicircle distributions

The family of power semicircle distributions defined as normal- ized real powers of the semicircle density is considered. The marginals of uniform distributions on spheres in high-dimensional Euclidean spaces are included in this family and a boundary case is the classical Gaussian dis- tribution. A review of some results including a genesis and the so-called Poincare's theorem is presented. The moments of these distributions are re- lated to the super Catalan numbers and their Cauchy transforms in terms of hypergeometric functions are derived. Some members of this class of dis- tributions play the role of the Gaussian distribution with respect to additive convolutions in non-commutative probability, such as the free, the monotone, the anti-monotone and the Boolean convolutions. The infinite divisibility of other power semicircle distributions with respect to these convolutions is stud- ied using simple kurtosis arguments. A connection between kurtosis and the free divisibility indicator is found. It is shown that for the classical Gaussian distribution the free divisibility indicator is strictly less than 2.

Vein Image Segmentation Based on Distribution Ratio of Directional Fields

In this paper,a novel segmentation algorithm is presented to improve segmentation performance for vein images.The proposed algorithm uses the distribution ratio of directional fields(DRDF) as the segmentation principle to distinguish vein lines from background.First,the neighbor information and template of neighbor segmentation are used to obtain directional fields for every pixel in the original image.Then in image of directional fields,DRDF of complementary semi-circle regions and distribution discrimination function are used to calculate the 8-grayscale image. Finally,a threshold is set to binarize the 8-grayscale image to obtain segmentation results.The algorithm combines the feature of vein image and spatial properties of directional fields image.Experiment results show that it is very effective for vein image with non-uniform illumination,inhomogeneous thickness,and blurry boundary.

Fixed trace β-Hermite ensembles: Asymptotic eigenvalue density and the edge of the density

In the present paper, fixed trace β-Hermite ensembles generalizing the fixed trace Gaussian ensembles are considered. For all β, we prove the Wigner semicircle law for these ensembles by using two different methods: one is the moment equivalence method with the help of the matrix model for general β, the other is to use asymptotic analysis tools. At the edge of the density, we prove that the edge scaling limit for β-HE implies the same limit for fixed trace β-Hermite ensembles. Consequently, explicit limit can be given for fixed trace Gaussian orthogonal, unitary, and symplectic ensembles. Furthermore, for even β, analogous to β-Hermite ensembles, a multiple integral of the Konstevich type can be obtained.

Bulk universality for Wigner matrices

AbstractWe consider N × N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density ν(x) = e−U(x). We prove that the eigenvalue statistics in the bulk are given by the Dyson sine kernel provided that U ∈ C6( \input amssym $\Bbb R$) with at most polynomially growing derivatives and ν(x) ≥ Ce−C|x| for x large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales. © 2010 Wiley Periodicals, Inc.

Near Resonance Charge Exchange in Ion−Atom Collisions of Lithium Isotopes

Collisions of ions and atoms of (6)Li and (7)Li are explored theoretically over a wide range of energy from 10(-14) to 1 eV. Accurate ab initio calculations are carried out of the Born-Oppenheimer potentials and the nonadiabatic couplings that are responsible for the near resonance charge exchange. Scattering studies show that the calculated charge exchange cross section follows Wigner's law for inelastic processes for energies below 10(-10) eV and that the zero temperature rate constant for it is 2.1 x 10(-9) cm(3) s(-1). At collision energies much larger than the isotope shift of the ionization potentials of the atoms, we show that the near resonance charge exchange process is equivalent to the resonance charge exchange with cross sections having a logarithmic dependence on energy. A comparison with the Langevin model at intermediate energies is also presented.

Wegner Estimate and Level Repulsion for Wigner Random Matrices

We consider N × N Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutive eigenvalues is of order 1/ N. Under suitable assumptions on the distribution of the single matrix element, we first prove that, away from the spectral edges, the empirical density of eigenvalues concentrates around the Wigner semicircle law on energy scales η ≫ N −1 . This result establishes the semicircle law on the optimal scale and it removes a logarithmic factor from our previous result [6]. We then show a Wegner estimate, i.e., that the averaged density of states is bounded. Finally, we prove that the eigenvalues of a Wigner matrix repel each other, in agreement with the universality conjecture.

Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices

We consider N×N Hermitian random matrices with i.i.d. entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. We study the connection between eigenvalue statistics on microscopic energy scales η≪1 and (de)localization properties of the eigenvectors. Under suitable assumptions on the distribution of the single matrix elements, we first give an upper bound on the density of states on short energy scales of order η∼log N/N. We then prove that the density of states concentrates around the Wigner semicircle law on energy scales η≫N−2/3. We show that most eigenvectors are fully delocalized in the sense that their ℓp-norms are comparable with N1/p−1/2 for p≥2, and we obtain the weaker bound N2/3(1/p−1/2) for all eigenvectors whose eigenvalues are separated away from the spectral edges. We also prove that, with a probability very close to one, no eigenvector can be localized. Finally, we give an optimal bound on the second moment of the Green function.

Eigenvalue separation in some random matrix models

The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semicircle law. If the Gaussian entries are all shifted by a constant amount s/(2N)1/2, where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semicircle provided s>1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an analogous comparative study of eigenvalue separation properties when the sizes of the matrices are fixed and s→∞, and higher rank analogs of this setting. This is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions and using the interpretation of the latter as a Green function in the Dyson Brownian motion model. For the shifted mean Gaussian unitary ensemble and its analogs, an alternative approach is to use exact expressions for the correlation functions in terms of classical orthogonal polynomials and associated multiple generalizations. By using these exact expressions to compute and plot the eigenvalue density, illustrations of the various eigenvalue separation effects are obtained.