Symmetry Properties of the k-Body Embedded Unitary Gaussian Ensemble of Random Matrices

We extend the recent study of the k-body embedded Gaussian ensembles by L. Benet, T. Rupp, and H. A. Weidenmüller (2001, Benet, Phys. Rev. Lett.87, 101601-1 and 2001, Ann. Phys. (N.Y.)292, 67) and by T. Asaga, L. Benet, T. Rupp, and H. A. Weidenmüller (cond-mat/0107363 and cond-mat/0107364). We show that central results of these papers can be derived directly from the symmetry properties of both the many-particle states and the random k-body interaction. We offer new insight into the structure of the matrix of second moments of the embedded ensemble and of the supersymmetry approach. We extend the concept of the embedded ensemble and define it purely group-theoretically.     

Edge Universality for Orthogonal Ensembles of Random Matrices

We prove edge universality of local eigenvalue statistics for orthogonal invariant matrix models with real analytic potentials and one interval limiting spectrum. Our starting point is the result of Shcherbina (Commun. Math. Phys. 285, 957–974, 2009) on the representation of the reproducing matrix kernels of orthogonal ensembles in terms of scalar reproducing kernel of corresponding unitary ensemble.     

On Universality for Orthogonal Ensembles of Random Matrices

We prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum. Our starting point is the Tracy-Widom formula for the matrix reproducing kernel. The key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a product of a positive Toeplitz matrix and a two diagonal skew symmetric Toeplitz matrix.     

On the 1/n Expansion for Some Unitary Invariant Ensembles of Random Matrices

We present a version of the 1/n-expansion for random matrix ensembles known as matrix models. The case where the support of the density of states of an ensemble consists of one interval and the case where the density of states is even and its support consists of two symmetric intervals is treated. In these cases we construct the expansion scheme for the Jacobi matrix determining a large class of expectations of symmetric functions of eigenvalues of random matrices, prove the asymptotic character of the scheme and give an explicit form of the first two terms. This allows us, in particular, to clarify certain theoretical physics results on the variance of the normalized traces of the resolvent of random matrices. We also find the asymptotic form of several related objects, such as smoothed squares of certain orthogonal polynomials, the normalized trace and the matrix elements of the resolvent of the Jacobi matrices, etc. Received: 9 November 2000 / Accepted: 26 July 2001     

Distribution Functions for Random Variables for Ensembles of Positive Hermitian Matrices

Distribution functions for random variables that depend on a parameter are computed asymptotically for ensembles of positive Hermitian matrices. The inverse Fourier transform of the distribution is shown to be a Fredholm determinant of a certain operator that is an analogue of a Wiener-Hopf operator. The asymptotic formula shows that, up to the terms of order o(1), the distributions are Gaussian. Received: 5 November 1996 / Accepted: 8 January 1997     

Lyapunov Exponents for Products of Complex Gaussian Random Matrices

The exact value of the Lyapunov exponents for the random matrix product P _N =A _N A _N?1?A ₁ with each $A_{i} = \varSigma^{1/2} G_{i}^{\mathrm{c}}$ , where Σ is a fixed d×d positive definite matrix and $G_{i}^{\mathrm{c}}$ a d×d complex Gaussian matrix with entries standard complex normals, are calculated. Also obtained is an exact expression for the sum of the Lyapunov exponents in both the complex and real cases, and the Lyapunov exponents for diffusing complex matrices.     

Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices

We study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble. We begin by considering an n×n matrix from the Gaussian orthogonal ensemble (GOE) or Gaussian symplectic ensemble (GSE) and let x _k denote eigenvalue number k. Under the condition that both k and n?k tend to infinity as n→∞, we show that x _k is normally distributed in the limit. We also consider the joint limit distribution of eigenvalues $(x_{k_{1}},\ldots,x_{k_{m}})$ from the GOE or GSE where k ₁, n?k _m and k _i+1?k _i , 1≤i≤m?1, tend to infinity with n. The result in each case is an m-dimensional normal distribution. Using a recent universality result by Tao and Vu, we extend our results to a class of Wigner real symmetric matrices with non-Gaussian entries that have an exponentially decaying distribution and whose first four moments match the Gaussian moments.     

An Estimate for the Average Spectral Measure of Random Band Matrices

For a class of random band matrices of band width W, we prove regularity of the average spectral measure at scales ϵ≥W ^−0.99, and find its asymptotics at these scales.     

Random Incidence Matrices: Moments of the Spectral Density

We study numerically and analytically the spectrum of incidence matrices of random labeled graphs on N vertices: any pair of vertices is connected by an edge with probability p. We give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in N and p. For large N and fixed p the spectrum contains a large eigenvalue at Np and a semicircle of small eigenvalues. For large N and fixed average connectivity pN (dilute or sparse random matrices limit) we show that the spectrum always contains a discrete component. An anomaly in the spectrum near eigenvalue 0 for connectivity close to e is observed. We develop recursion relations to compute the moments as explicit polynomials in pN. Their growth is slow enough so that they determine the spectrum. The extension of our methods to the Laplacian matrix is given in Appendix.     

Gram Matrices of Mixed-State Ensembles

International Journal of Theoretical Physics - Gram matrices arise naturally in the consideration of pair-wise overlap of a family of vectors or quantum pure states, and play an important role in...     

Spectral Properties of Amphiflavins in Polymer Matrices

The parameters characterizing the absorption, fluorescence and phosphorescence of 10-dodecylisoalloxazine (DIA), 10-octadecylisoalloxazine (OIA) 3-methyl-10-dodecylisoalloxazine (MDIA) embedded into polymeric matrices (polystyrene – PS, polymethacrylate methyl – PMM and poly(vinyl alcohol) – PVA) were studied. It was found that both the polarity of microenvironment and its structure influenced the changes of the spectral properties of the examined amphiflavins.     

Some Universal Properties for Restricted Trace Gaussian Orthogonal,Unitary and Symplectic Ensembles

Consider fixed and bounded trace Gaussian orthogonal, unitary and symplectic ensembles, closely related to Gaussian ensembles without any constraint. For three restricted trace Gaussian ensembles, we prove universal limits of correlation functions at zero and at the edge of the spectrum edge. Our argument also applies to restricted trace ensembles with monomial potentials. In addition, by using the universal result in the bulk for fixed trace Gaussian unitary ensemble, which has been obtained by Götze and Gordin, we also prove the universal limits of correlation functions everywhere in the bulk for bounded trace Gaussian unitary ensemble.     

Spectral Measure of Heavy Tailed Band and Covariance Random Matrices

We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N × N symmetric matrix whose (i, j) entry is , where (x _ij , 1 ≤ i ≤ j < ∞) is an infinite array of i.i.d real variables with common distribution in the domain of attraction of an α-stable law, , and σ is a deterministic function. For random diagonal D _N independent of and with appropriate rescaling a _N , we prove that converges in mean towards a limiting probability measure which we characterize. As a special case, we derive and analyze the almost sure limiting spectral density for empirical covariance matrices with heavy tailed entries. Supported in part by a Discovery grant from the Natural Sciences and Engineering Research Council of Canada and a University of Saskatchewan start-up grant. Research partially supported by NSF grant #DMS-0806211.     

On Asymptotic Expansions and Scales of Spectral Universality in Band Random Matrix Ensembles

 We consider real random symmetric N × N matrices H of the band-type form with characteristic length b. The matrix entries are independent Gaussian random variables and have the variance proportional to , where u(t) vanishes at infinity. We study the resolvent in the limit and obtain the explicit expression for the leading term of the first correlation function of the normalized trace . We examine on the local scale and show that its asymptotic behavior is determined by the rate of decay of u(t). In particular, if u(t) decays exponentially, then . This expression is universal in the sense that the particular form of u determines the value of C > 0 only. Our results agree with those detected in both numerical and theoretical physics studies of spectra of band random matrices. Received: 8 April 2000 / Accepted: 7 June 2002 Published online: 21 October 2002 RID="*" ID="*" Present address: Département de Mathématiques, Université de Versailles Saint-Quentin, 78035 Versailles, France.     

The Optical Properties of Polydisperse Glass Microsphere Ensembles in the Infrared Spectral Region

Semitransparent composites with glass microspheres are promising modern materials. In particular, they contribute to the increase of the efficiency of heat-insulating coatings. An experimental study of the transmission and scattering of infrared radiation (at wavelengths from 2 to 25 μm) by a sodium borosilicate glass microsphere powder is presented. The spectra, angular-scattering patterns, and the degree of polarization of scattered radiation were studied.     

Convergence Rates of the Spectral Distributions of Large Random Quaternion Self-Dual Hermitian Matrices

In this paper, convergence rates of the spectral distributions of quaternion self-dual Hermitian matrices are investigated. We show that under conditions of finite 6th moments, the expected spectral distribution of a large quaternion self-dual Hermitian matrix converges to the semicircular law in a rate of \(O(n^{-1/2})\) and the spectral distribution itself converges to the semicircular law in rates \(O_p(n^{-2/5})\) and \(O_{a.s.}(n^{-2/5+\eta })\) . Those results include GSE as a special case.     

Large n Limit of Gaussian Random Matrices with External Source, Part I

We consider the random matrix ensemble with an external sourcedefined on n×n Hermitian matrices, where A is a diagonal matrix with only two eigenvalues ±a of equal multiplicity. For the case a>1, we establish the universal behavior of local eigenvalue correlations in the limit n, which is known from unitarily invariant random matrix models. Thus, local eigenvalue correlations are expressed in terms of the sine kernel in the bulk and in terms of the Airy kernel at the edge of the spectrum. We use a characterization of the associated multiple Hermite polynomials by a 3×3-matrix Riemann-Hilbert problem, and the Deift/Zhou steepest descent method to analyze the Riemann-Hilbert problem in the large n limit.Dedicated to Freeman Dyson on his eightieth birthdayThe first author was supported in part by NSF Grants DMS-9970625 and DMS-0354962.The second author was supported in part by projects G.0176.02 and G.0455.04 of FWO-Flanders, by K.U.Leuven research grant OT/04/24, and by INTAS Research Network NeCCA 03-51-6637.     

Entropy and the Shannon-McMillan-Breiman Theorem for Beta Random Matrix Ensembles

We show that beta ensembles in Random Matrix Theory with generic real analytic potential have the asymptotic equipartition property. In addition, we prove a Central Limit Theorem for the density of the eigenvalues of these ensembles.