Random block matrices generalizing the classical Jacobi and Laguerre ensembles

AbstractIn this paper we consider random block matrices which generalize the classical Laguerre ensemble and the Jacobi ensemble. We show that the random eigenvalues of the matrices can be uniformly approximated by the zeros of matrix orthogonal polynomials and obtain a rate for the maximum difference between the eigenvalues and the zeros. This relation between the random block matrices and matrix orthogonal polynomials allows a derivation of the asymptotic spectral distribution of the matrices

The generalized Cartan decomposition for classical random matrix ensembles

We present a completed classification of the classical random matrix ensembles: Hermite (Gaussian), Laguerre (Wishart), Jacobi (MANOVA) and Circular by introducing the concept of the generalized Cartan decomposition and the double coset space. Previous authors associate a symmetric space $G/K$ with a random matrix density on the double coset structure $K\backslash G/K$. However this is incomplete. Complete coverage requires the double coset structure $A = K_1\backslash G/K_2$, where $G/K_1$ and $G/K_2$ are two symmetric spaces. Furthermore, we show how the matrix factorization obtained by the generalized Cartan decomposition $G = K_1AK_2$ plays a crucial role in sampling algorithms and the derivation of the joint probability density of $A$.Comment: 26 page

The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensembles with a source

In classical random matrix theory the Gaussian and chiral Gaussian random matrix models with a source are realized as shifted mean Gaussian, and chiral Gaussian, random matrices with real $(\beta = 1)$, complex ($\beta = 2)$ and real quaternion $(\beta = 4$) elements. We use the Dyson Brownian motion model to give a meaning for general $\beta > 0$. In the Gaussian case a further construction valid for $\beta > 0$ is given, as the eigenvalue PDF of a recursively defined random matrix ensemble. In the case of real or complex elements, a combinatorial argument is used to compute the averaged characteristic polynomial. The resulting functional forms are shown to be a special cases of duality formulas due to Desrosiers. New derivations of the general case of Desrosiers' dualities are given. A soft edge scaling limit of the averaged characteristic polynomial is identified, and an explicit evaluation in terms of so-called incomplete Airy functions is obtained.Comment: 21 page

From Wishart to Jacobi ensembles: Statistical properties and applications

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Sixty years after the works of Wigner and Dyson, Random Matrix Theory still remains a very active and challenging area of research, with countless applications in mathematical physics, statistical mechanics and beyond. In this thesis, we focus on rotationally invariant models where the requirement of independence of matrix elements is dropped. Some classical examples are the Jacobi and Wishart-Laguerre (or chiral) ensembles, which constitute the core of the present work. The Wishart-Laguerre ensemble contains covariance matrices of random data, and represents a very important tool in multivariate data analysis, with recent applications to finance and telecommunications. We will first consider large deviations of the maximum eigenvalue, providing new analytical results for its large N behavior, and then a power-law deformation of the classical Wishart-Laguerre ensemble, with possible applications to covariance matrices of financial data. For the Jacobi matrices, which arise naturally in the quantum conductance problem, we provide analytical formulas for quantities of interest for the experiments

The stochastic operator approach to random matrix theory

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 147-150) and index.Classical random matrix models are formed from dense matrices with Gaussian entries. Their eigenvalues have features that have been observed in combinatorics, statistical mechanics, quantum mechanics, and even the zeros of the Riemann zeta function. However, their eigenvectors are Haar-distributed-completely random. Therefore, these classical random matrices are rarely considered as operators. The stochastic operator approach to random matrix theory, introduced here, shows that it is actually quite natural and quite useful to view random matrices as random operators. The first step is to perform a change of basis, replacing the traditional Gaussian random matrix models by carefully chosen distributions on structured, e.g., tridiagonal, matrices. These structured random matrix models were introduced by Dumitriu and Edelman, and of course have the same eigenvalue distributions as the classical models, since they are equivalent up to similarity transformation. This dissertation shows that these structured random matrix models, appropriately rescaled, are finite difference approximations to stochastic differential operators. Specifically, as the size of one of these matrices approaches infinity, it looks more and more like an operator constructed from either the Airy operator, ..., or one of the Bessel operators, ..., plus noise. One of the major advantages to the stochastic operator approach is a new method for working in "general [beta] " random matrix theory. In the stochastic operator approach, there is always a parameter [beta] which is inversely proportional to the variance of the noise.(cont.) In contrast, the traditional Gaussian random matrix models identify the parameter [beta] with the real dimension of the division algebra of elements, limiting much study to the cases [beta] = 1 (real entries), [beta] = 2 (complex entries), and [beta] = 4 (quaternion entries). An application to general [beta] random matrix theory is presented, specifically regarding the universal largest eigenvalue distributions. In the cases [beta] = 1, 2, 4, Tracy and Widom derived exact formulas for these distributions. However, little is known about the general [beta] case. In this dissertation, the stochastic operator approach is used to derive a new asymptotic expansion for the mean, valid near [beta] = [infinity]. The expression is built from the eigendecomposition of the Airy operator, suggesting the intrinsic role of differential operators. This dissertation also introduces a new matrix model for the Jacobi ensemble, solving a problem posed by Dumitriu and Edelman, and enabling the extension of the stochastic operator approach to the Jacobi case.by Brian D. Sutton.Ph.D