Ioana Dumitriu

Matrix models for beta ensembles

This paper constructs tridiagonal random matrix models for general (β>0) β-Hermite (Gaussian) and β-Laguerre (Wishart) ensembles. These generalize the well-known Gaussian and Wishart models for β=1,2,4. Furthermore, in the cases of the β-Laguerre ensembles, we eliminate the exponent quantization present in the previously known models. We further discuss applications for the new matrix models, and present some open problems.

Global spectrum fluctuations for the β-Hermite and β-Laguerre ensembles via matrix models

We study the global spectrum fluctuations for β-Hermite and β-Laguerre ensembles via the tridiagonal matrix models introduced previously by the present authors [J. Math. Phys. 43, 5830 (2002)], and prove that the fluctuations describe a Gaussian process on polynomials. We extend our results to slightly larger classes of random matrices.

Sparse regular random graphs: Spectral density and eigenvectors

We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to the semicircle law. Moreover, we prove concentration estimates on the number of eigenvalues over progressively smaller intervals. We also show that, with high probability, all the eigenvectors are delocalized.

MOPS: Multivariate orthogonal polynomials (symbolically)

In this paper we present a Maple library (MOPS) for computing Jack, Hermite, Laguerre, and Jacobi multivariate polynomials, as well as eigenvalue statistics for the Hermite, Laguerre, and Jacobi ensembles of random matrix theory. We also compute multivariate hypergeometric functions, and offer both symbolic and numerical evaluations for all these quantities. We prove that all algorithms are well-defined, analyze their complexity, and illustrate their performance in practice. Finally, we present a few applications of this library.

Fast matrix multiplication is stable

We perform forward error analysis for a large class of recursive matrix multiplication algorithms in the spirit of Bini and Lotti [Numer. Math. 36:63–72, 1980]. As a consequence of our analysis, we show that the exponent of matrix multiplication (the optimal running time) can be achieved by numerically stable algorithms. We also show that new group-theoretic algorithms proposed in Cohn and Umans [Foundations of Computer Science, 44th Annual IEEE Symposium, pp. 438–449, 2003] and Cohn et al. [Foundations of Computer Science, 46th Annual IEEE Symposium, pp. 379–388, 2005] are all included in the class of algorithms to which our analysis applies, and are therefore numerically stable. We perform detailed error analysis for three specific fast group-theoretic algorithms.

Eigenvalues of Hermite and Laguerre ensembles: large beta asymptotics

In this paper we examine the zero and first order eigenvalue fluctuations for the β-Hermite and β-Laguerre ensembles, using tridiagonal matrix models, in the limit as β→∞. We prove that the fluctuations are described by multivariate Gaussians of covariance O(1/β), centered at the roots of a corresponding Hermite (Laguerre) polynomial. The covariance matrix itself is expressed as combinations of Hermite or Laguerre polynomials respectively. We show that the approximations are of real value even for small β; we can use them to approximate the true functions even for the traditional β=1,2,4 values.

Distributions of the Extreme Eigenvaluesof Beta-Jacobi Random Matrices

We present explicit formulas for the distributions of the extreme eigenvalues of the

The Liar Game Over an Arbitrary Channel

\beta

Functional limit theorems for random regular graphs

-Jacobi random matrix ensemble in terms of the hypergeometric function of a matrix argument. For

Accurate and Efficient Expression Evaluation and Linear Algebra

\beta=1,2,4,