Canonical Moments and Random Spectral Measures

We study some connections between the random moment problem and random matrix theory. A uniform draw in a space of moments can be lifted into the spectral probability measure of the pair (A,e), where A is a random matrix from a classical ensemble, and e is a fixed unit vector. This random measure is a weighted sampling among the eigenvalues of A. We also study the large deviations properties of this random measure when the dimension of the matrix increases. The rate function for these large deviations involves the reversed Kullback information.     

Tridiagonal Random Matrix: Gaussian Fluctuations and Deviations

This paper is devoted to the Gaussian fluctuations and deviations of the traces of tridiagonal random matrices. Under quite general assumptions, we prove that the traces are approximately normally distributed. A Multi-dimensional central limit theorem is also obtained here. These results have several applications to various physical models and random matrix models, such as the Anderson model, the random birth–death Markov kernel, the random birth–death Q matrix and the \(\beta \)-Hermite ensemble. Furthermore, under an independent-and-identically-distributed condition, we also prove the large deviation principle as well as the moderate deviation principle for the traces.     

Meixner Matrix Ensembles

We construct a family of matrix ensembles that fits Anshelevich’s regression postulates for “Meixner laws on matrices,” namely the distribution with an invariance property of X when $\mathbb{E}(\mathbf {X}^{2}|\mathbf {X}+\mathbf {Y})=a(\mathbf {X}+\mathbf {Y})^{2}+b(\mathbf {X}+\mathbf {Y})+c\mathbf {I}_{n}$ where X and Y are i.i.d. symmetric matrices of order n. We show that the Laplace transform of a general n×n Meixner matrix ensemble satisfies a system of partial differential equations which is explicitly solvable for n=2. We rely on these solutions to identify the six types of 2×2 Meixner matrix ensembles.     

A Simple Approach to the Global Regime of Gaussian Ensembles of Random Matrices

We present simple proofs of several basic facts of the global regime (the existence and the form of the nonrandom limiting Normalized Counting Measure of eigenvalues, and the central limit theorem for the trace of the resolvent) for ensembles of random matrices whose probability law involves the Gaussian distribution. The main difference with previous proofs is the systematic use of the Poincare-Nash inequality, allowing us to obtain the O(n ⁻²) bounds for the variance of the normalized trace of the resolvent that are valid up to the real axis in the spectral parameter. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 6, pp. 790–817, June, 2005.     

Hermitean Matrix Ensembles and Orthogonal Polynomials

In this article, we investigate orthogonal polynomials associated with complex Hermitean matrix ensembles using the combination of the methods of Coulomb fluid (or potential theory), chain sequences, and Birkhoff–Trjitzinsky theory. We give a general formula for the largest eigenvalue of the N×N Jacobi matrices (which is equivalent to estimating the largest zero of a sequence of orthogonal polynomials) and the two-level correlation function for the α ensembles (α>0) introduced previously for α>1. In the case of 0<α<1, we give a natural representation for the weight function that is a special case of the general Nevanlinna parametrization. We also discuss Hermitean matrix ensembles associated with general indeterminate moment problems.     

Finite Size Effects for Spacing Distributions in Random Matrix Theory: Circular Ensembles and Riemann Zeros

According to Dyson's threefold way, from the viewpoint of global time reversal symmetry, there are three circular ensembles of unitary random matrices relevant to the study of chaotic spectra in quantum mechanics. These are the circular orthogonal, unitary, and symplectic ensembles, denoted COE, CUE, and CSE, respectively. For each of these three ensembles and their thinned versions, whereby each eigenvalue is deleted independently with probability , we take up the problem of calculating the first two terms in the scaled large N expansion of the spacing distributions. It is well known that the leading term admits a characterization in terms of both Fredholm determinants and Painlevé transcendents. We show that modifications of these characterizations also remain valid for the next to leading term, and that they provide schemes for high precision numerical computations. In the case of the CUE, there is an application to the analysis of Odlyzko's data set for the Riemann zeros, and in that case, some further statistics are similarly analyzed.     

Hausdorff-type Measures of the Sample Path of Gaussian Random Fields

Let φ be a Hausdorff measure function and A be an infinite increasing sequence of positive integers. The Hausdorff-type measure φ - mA associated to φ and A is studied. Let X（t）（t ∈ R^N） be certain Gaussian random fields in R^d. We give the exact Hausdorff measure of the graph set GrX（[0, 1]N）, and evaluate the exact φ - mA measure of the image and graph set of X（t）. A necessary and sufficient condition on the sequence A is given so that the usual Hausdorff measure function for X（[0, 1] ^N） and GrX（[0, 1]^N） are still the correct measure functions. If the sequence A increases faster, then some smaller measure functions will give positive and finite （ φ A）-Hausdorff measure for X（[0, 1]^N） and GrX（[0, 1]N）.     

Moments of Markov Random Evolutions

We find moments of a process of Markov random evolutions in a finite-dimensional space.     

Modified Moments and Matrix Orthogonal Polynomials

In the scalar case, computation of recurrence coefficients of polynomials orthogonal with respect to a nonnegative measure is done via the modified Chebyshev algorithm. Using the concept of matrix biorthogonality, we extend this algorithm to the vector case.     

Small Ball Probabilities Around Random Centers of Gaussian Measures and Applications to Quantization

Let be a centered Gaussian measure on a separable Hilbert space (E, ). We are concerned with the logarithmic small ball probabilities around a -distributed center X. It turns out that the asymptotic behavior of –log (B(X,)) is a.s. equivalent to that of a deterministic function _R (). These new insights will be used to derive the precise asymptotics of a random quantization problem which was introduced in a former article by Dereich, Fehringer, Matoussi, and Scheutzow.⁽⁸⁾     

The Gaussian Moments Conjecture and the Jacobian Conjecture

We first propose what we call the Gaussian Moments Conjecture. We then show that the Jacobian Conjecture follows from the Gaussian Moments Conjecture. Note that the the Gaussian Moments Conjecture is a special case of [11, Conjecture 3.2]. The latter conjecture was referred to as the Moment Vanishing Conjecture in [7, Conjecture A] and the Integral Conjecture in [6, Conjecture 3.1] (for the one-dimensional case). We also give a counter-example to show that [11, Conjecture 3.2] fails in general for polynomials in more than two variables.     

正随机向量的矩

本文给出一个计算正随机向量矩的统一的方法，并对一些特殊的重要的分布类具体给出矩的表达式     

Ising Chain with Random Magnetic Moments

The Ising chain with random magnetic moments is discussed, and the thermodynamic functions are computed.     

The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law

LetAbe annbynmatrix whose elements are independent random variables with standard normal distributions. Girko's (more general) circular law states that the distribution of appropriately normalized eigenvalues is asymptotically uniform in the unit disk in the complex plane. We derive the exact expected empirical spectral distribution of the complex eigenvalues for finiten, from which convergence in the expected distribution to the circular law for normally distributed matrices may be derived. Similar methodology allows us to derive a joint distribution formula for the real Schur decomposition ofA. Integration of this distribution yields the probability thatAhas exactlykreal eigenvalues. For example, we show that the probability thatAhas all real eigenvalues is exactly 2^{−n(n−1)/4}.     

Average Approximations and Moments of Measures

We investigate average approximations of infinite dimensional mappings and related problems connected with moments of measures on linear spaces. A conjecture stated by J. F. Traub and A. G. Werschulz (1994, Math. Intelligencer16, 42–48) is settled. Several positive results concerning average approximations of Banach space valued mappings are obtained. Some related open problems are discussed.     

Smoothing Windows for the Synthesis of Gaussian Stationary Random Fields Using Circulant Matrix Embedding

When generating Gaussian stationary random fields, a standard method based on circulant matrix embedding usually fails because some of the associated eigenvalues are negative. The eigenvalues can be shown to be nonnegative in the limit of increasing sample size. Computationally feasible large sample sizes, however, rarely lead to nonnegative eigenvalues. Another solution is to extend suitably the covariance function of interest so that the eigenvalues of the embedded circulant matrix become nonnegative in theory. Though such extensions have been found for a number of examples of stationary fields, the method depends on nontrivial constructions in specific cases.

In this work, the embedded circulant matrix is smoothed at the boundary by using a cutoff window or overlapping windows over a transition region. The windows are not specific to particular examples of stationary fields. The resulting method modifies the standard circulant embedding, and is easy to use. It is shown that this straightforward approach works for many examples of interest, with the overlapping windows performing consistently better. The method even outperforms in the cases where extending the covariance leads to nonnegative eigenvalues in theory, in the sense that the transition region is considerably smaller. The Matlab code implementing the method is included in the online supplementary materials and also publicly available at www.hermir.org.     

Curvature Measures and Random Sets,I

The paper deals with signed curvature measures as introduced by Federer for sets with positive reach. An integral representation and a local Steiner formula for these measures are given. The main result is the additive extension of the curvature measures to locally finite unions of compatible sets with positive reach. Within this comprehensive class of subsets of R^d a generalized Steiner polynomial (local version) and section theorems (principal kinematic formula, Crofton formula) for the curvature measures are derived.