Non-Hermitian Tridiagonal Random Matrices and Returns to the Origin of a Random Walk

We study a class of tridiagonal matrix models, the q-roots of unity models, which includes the sign (q=2) and the clock (q=) models by Feinberg and Zee. We find that the eigenvalue densities are bounded by and have the symmetries of the regular polygon with 2q sides, in the complex plane. Furthermore, the averaged traces of M ^k are integers that count closed random walks on the line such that each site is visited a number of times multiple of q. We obtain an explicit evaluation for them.     

From Random Matrices to Stochastic Operators

We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonly arise, each associated with a familiar class of local eigenvalue behavior. The stochastic Airy operator displays soft edge behavior, associated with the Airy kernel. The stochastic Bessel operator displays hard edge behavior, associated with the Bessel kernel. The article concludes with suggestions for a stochastic sine operator, which would display bulk behavior, associated with the sine kernel.     

On the Law of Multiplication of Random Matrices

We recover Voiculescu's results on multiplicative free convolutions of probability measures by different techniques which were already developed by Pastur and Vasilchuk for the law of addition of random matrices. Namely, we study the normalized eigenvalue counting measure of the product of two n×n unitary matrices and the measure of the product of three n×n Hermitian (or real symmetric) positive matrices rotated independently by random unitary (or orthogonal) Haar distributed matrices. We establish the convergence in probability as n to a limiting nonrandom measure and obtain functional equations for the Herglotz and Stieltjes transforms of that limiting measure.     

Wigner Random Matrices with Non-Symmetrically Distributed Entries

We show that the spectral radius of an N× N random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from above by 2 σ + o(N−6/11+ε), where σ² is the variance of the matrix entries and ε is an arbitrary small positive number. Our bound improves the earlier results by Z. Füredi and J. Komlós (1981), and Van Vu (2005).     

Classical Skew Orthogonal Polynomials and Random Matrices

Skew orthogonal polynomials arise in the calculation of the n-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials. In the case that the eigenvalue probability density function involves a classical weight function, explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials, and the structure is used to give a closed-form expression for the sum. This theory treates all classical cases on an equal footing, giving formulas applicable at once to the Hermite, Laguerre, and Jacobi cases.     

Perturbation of Transmission Matrices in Nonlinear Random Media

Random media with tailored optical properties are attracting burgeoning interest for applications in imaging, biophysics, energy, nanomedicine, spectroscopy, cryptography, and telecommunications. A key paradigm for devices based on this class of materials is the transmission matrix, the tensorial link between the input and the output signals, that describes in full their optical behavior. The transmission matrix has specific statistical properties, such as the existence of lossless channels, that can be used to transmit information, and are determined by the disorder distribution. In nonlinear materials, these channels may be modulated and the transmission matrix tuned accordingly. Here, the direct measurement of the nonlinear transmission matrix of complex materials is reported, exploiting the strong optothermal nonlinearity of scattering silica aerogel (SA). It is shown that the dephasing effects due to nonlinearity are both controllable and reversible, opening the road to applications based on the nonlinear response of random media.     

Determinants of Airy Operators and Applications to Random Matrices

The purpose of this paper is to describe asymptotic formulas for determinants of certain operators that are analogues of Wiener–Hopf operators. The determinant formulas yield information about the distribution functions for certain random variables that arise in random matrix theory when one rescales at the edge of the spectrum.     

Power law scaling of the top Lyapunov exponent of a Product of Random Matrices

A sequence of i.i.d. matrix-valued random variables with probabilityp and with probability 1–p is considered. Leta() = a ₀ + O(), c() = c ₀ + O() lim ₀ b() = Oa ₀,c ₀, >0, andb()>0 for all >0. It is shown show that the top Lyapunov exponent of the matrix productX _n X _n-1...X ₁, = lim_n (1/n) n X _n X _n-1...X ₁ satisfies a power law with an exponent 1/2. That is, lim ₀(ln /ln ) = 1/2.     

Continuity of the Integrated Density of States on Random Length Metric Graphs

We establish several properties of the integrated density of states for random quantum graphs: Under appropriate ergodicity and amenability assumptions, the integrated density of states can be defined using an exhaustion procedure by compact subgraphs. A trace per unit volume formula holds, similarly as in the Euclidean case. Our setting includes periodic graphs. For a model where the edge lengths are random and vary independently in a smooth way we prove a Wegner estimate and related regularity results for the integrated density of states. These results are illustrated for an example based on the Kagome lattice. In the periodic case we characterise all compactly supported eigenfunctions and calculate the position and size of discontinuities of the integrated density of states.      

On the Distribution of the Information Density of Gaussian Random Vectors: Explicit Formulas and Tight Approximations

Based on the canonical correlation analysis, we derive series representations of the probability density function (PDF) and the cumulative distribution function (CDF) of the information density of arbitrary Gaussian random vectors as well as a general formula to calculate the central moments. Using the general results, we give closed-form expressions of the PDF and CDF and explicit formulas of the central moments for important special cases. Furthermore, we derive recurrence formulas and tight approximations of the general series representations, which allow efficient numerical calculations with an arbitrarily high accuracy as demonstrated with an implementation in Python publicly available on GitLab. Finally, we discuss the (in)validity of Gaussian approximations of the information density.     

Spectral Moments of the Long-term Average Spectrum: Sensitive Indices of Voice Change After Therapy?

Voice clinicians require an objective, reliable, and relatively automatic method to assess voice change after medical, surgical, or behavioral intervention. This measure must be sensitive to a variety of voice qualities and severities, and preferably should reflect voice in continuous speech. The long-term average spectrum (LTAS) is a fast Fourier transform-generated power spectrum whose properties can be compared with a Gaussian bell curve using spectral moments analysis. Four spectral moments describe features of the LTAS: Spectral mean (Moment 1) and standard deviation (Moment 2) represent the spectrum's central tendency and dispersion, respectively. Skewness (based on Moment 3) and kurtosis (based on Moment 4) represent the spectrum's tilt and peakedness, respectively. To examine whether the first four spectral moments of the LTAS were sensitive to perceived voice improvement after voice therapy, this investigation compared pretreatment and posttreatment voice samples of 93 patients with functional dysphonia using spectral moments analysis. Inspection of the results revealed that spectral mean and standard deviation lowered significantly with perceived voice improvement after successful behavioral management (p < 0.001). However, changes in skewness and kurtosis were not significant. Furthermore, lowering of the spectral mean uniquely accounted for approximately 14% of the variance in the pretreatment to posttreatment changes observed in perceptual ratings of voice severity (p < 0.001), indicating that spectral mean (ie, Moment 1) of the LTAS may be one acoustic marker sensitive to improvement in dysphonia severity.     

Band Edge Behavior of the Integrated Density of States of Random Jacobi Matrices in Dimension 1

Let H be a Jacobi matrix acting on and V a random potential of Anderson type. Let H = H+V . We give a general formula relating the decay of the integrated density of states of H at the edges of the almost sure spectrum of H to the decay of the integrated density of states of H at the edges of the spectrum of H.     

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.     

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.     

Asymptotic Distribution of Eigenvalues of Weakly Dilute Wishart Matrices

We study the eigenvalue distribution of large random matrices that are randomly diluted. We consider two random matrix ensembles that in the pure (nondilute) case have a limiting eigenvalue distribution with a singular component at the origin. These include the Wishart random matrix ensemble and Gaussian random matrices with correlated entries. Our results show that the singularity in the eigenvalue distribution is rather unstable under dilution and that even weak dilution destroys it. This revised version was published online in July 2006 with corrections to the Cover Date.     

Elementary Proof for Asymptotics of Large Haar-Distributed Unitary Matrices

We provide an elementary proof for a theorem due to Petz and Réffy which states that for a random n × n unitary matrix with distribution given by the Haar measure on the unitary group U(n), the upper left (or any other) k × k submatrix converges in distribution, after multiplying by a normalization factor and as , to a matrix of independent complex Gaussian random variables with mean 0 and variance 1.      

A Note on the Eigenvalue Density of Random Matrices

The distribution of eigenvalues of N 2 N random matrices in the limit N M X is the solution to a variational principle that determines the ground state energy of a confined fluid of classical unit charges. This fact is a consequence of a more general theorem, proven here, in the statistical mechanics of unstable interactions. Our result establishes the eigenvalue density of some ensembles of random matrices which were not covered by previous theorems.     

On Asymptotic Behavior of Multilinear Eigenvalue Statistics of Random Matrices

We prove the Law of Large Numbers and the Central Limit Theorem for analogs of U- and V- (von Mises) statistics of eigenvalues of random matrices as their size tends to infinity. We show first that for a certain class of test functions (kernels), determining the statistics, the validity of these limiting laws reduces to the validity of analogous facts for certain linear eigenvalue statistics. We then check the conditions of the reduction statements for several most known ensembles of random matrices. The reduction phenomenon is well known in statistics, dealing with i.i.d. random variables. It is of interest that an analogous phenomenon is also the case for random matrices, whose eigenvalues are strongly dependent even if the entries of matrices are independent.     

Gaussian-Random Ensembles of Pseudo-Hermitian Matrices

Attention has been brought to the possibility that statistical fluctuation properties of several complex spectra, or well-known number sequences, may display strong signatures that the Hamiltonian yielding them as eigenvalues is PT-symmetric (pseudo-Hermitian). We find that the random matrix theory of pseudo-Hermitian Hamiltonians gives rise to new universalities of level-spacing distributions other than those of GOE, GUE and GSE of Wigner and Dyson. We call the new proposals as Gaussian Pseudo-Orthogonal Ensemble and Gaussian Pseudo-Unitary Ensemble. We are also led to speculate that the enigmatic Riemann-zeros (1/2±it _n would rather correspond to some PT-symmetric (pseudo-Hermitian) Hamiltonian.