Arno B. J. Kuijlaars

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

AbstractWe consider the random matrix ensemble with an external source defined 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.

Random matrices with external source and multiple orthogonal polynomials

We show that the average characteristic polynomial Pn(z) = E[det(zI−M)] of the random Hermitian matrix ensemble Z 1 n exp(−Tr(V (M) − AM))dM is characterized by multiple orthogonality conditions that depend on the eigenvalues of the external source A. For each eigenvalue aj of A, there is a weight and Pn has nj orthogonality conditions with respect to this weight, if nj is the multiplicity of aj. The eigenvalue correlation functions have determinantal form, as shown by Zinn-Justin. Here we give a different expression for the kernel. We derive a Christoffel-Darboux formula in case A has two distinct eigenvalues, which leads to a compact formula in terms of a Riemann-Hilbert problem that is satisfied by multiple orthogonal polynomials.

Riemann-Hilbert Problems for Multiple Orthogonal Polynomials

In the early nineties, Fokas, Its and Kitaev observed that there is a natural Riemann-Hilbert problem (for 2 x×2 matrix functions) associated with a system of orthogonal polynomials. This Riemann-Hilbert problem was later used by Deift et al. and Bleher and Its to obtain interesting results on orthogonal polynomials, in particular strong asymptotics which hold uniformly in the complex plane. In this paper we will show that a similar Riemann-Hilbert problem (for (r + 1) × (r + 1) matrix functions) is associated with multiple orthogonal polynomials. We show how this helps in understanding the relation between two types of multiple orthogonal polynomials and the higher order recurrence relations for these polynomials. Finally we indicate how an extremal problem for vector potentials is important for the normalization of the Riemann-Hilbert problem. This extremal problem also describes the zero behavior of the multiple orthogonal polynomials.

Universality for eigenvalue correlations from the modified Jacobi unitary ensemble

The eigenvalue correlations of random matrices from the Jacobi Unitary Ensemble have a known asymptotic behavior as their size tends to infinity. In the bulk of the spectrum the behavior is described in terms of the sine kernel, and at the edge in terms of the Bessel kernel. We will prove that this behavior persists for the Modified Jacobi Unitary Ensemble. This generalization of the Jacobi Unitary Ensemble is associated with the modified Jacobi weight w(x)=(1-x)^\alpha (1+x)^\beta h(x) where the extra factor h is assumed to be real analytic and strictly positive on [-1,1]. We use the connection with the orthogonal polynomials with respect to the modified Jacobi weight, and recent results on strong asymptotics derived by K.T-R McLaughlin, W. Van Assche and the authors.

Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields

The equilibrium measure in the presence of an external field plays a role in a number of areas in analysis, for example, in random matrix theory: The limiting mean density of eigenvalues is precisely the density of the equilibrium measure. Typical behavior for the equilibrium measure is: 1 it is positive on the interior of a finite number of intervals, 2 it vanishes like a square root at endpoints, and 3 outside the support, there is strict inequality in the Euler-Lagrange variational conditions. If these conditions hold, then the limiting local eigenvalue statistics is loosely described by a “bulk,” in which there is universal behavior involving the sine kernel, and “edge effects,” in which there is a universal behavior involving the Airy kernel. Through techniques from potential theory and integrable systems, we show that this “regular” behavior is generic for equilibrium measures associated with real analytic external fields. In particular, we show that for any one-parameter family of external fields V/c, the equilibrium measure exhibits this regular behavior except for an at most countable number of values of c. We discuss applications of our results to random matrices, orthogonal polynomials, and integrable systems.

Impossibility of Fast Stable Approximation of Analytic Functions from Equispaced Samples

It is shown that no stable procedure for approximating functions from equally spaced samples can converge exponentially for analytic functions. To avoid instability, one must settle for root-exponential convergence. The proof combines a Bernstein inequality of 1912 with an estimate due to Coppersmith and Rivlin in 1992.

Universality for Eigenvalue Correlations at the Origin of the Spectrum

AbstractWe establish universality of local eigenvalue correlations in unitary random matrix ensembles

Multiple orthogonal polynomials of mixed type and non-intersecting Brownian motions

{{\frac{{1}}{{Z_n}}|\det M|^{{2\alpha}} e^{{-n{{\rm{ tr}}}\, V(M)}} dM}}

Convergence Analysis of Krylov Subspace Iterations with Methods from Potential Theory

near the origin of the spectrum. If V is even, and if the recurrence coefficients of the orthogonal polynomials associated with |x|2αe−nV(x) have a regular limiting behavior, then it is known from work of Akemann et al., and Kanzieper and Freilikher that the local eigenvalue correlations have universal behavior described in terms of Bessel functions. We extend this to a much wider class of confining potentials V. Our approach is based on the steepest descent method of Deift and Zhou for the asymptotic analysis of Riemann-Hilbert problems. This method was used by Deift et al. to establish universality in the bulk of the spectrum. A main part of the present work is devoted to the analysis of a local Riemann-Hilbert problem near the origin.

Strong asymptotics for Jacobi polynomials with varying nonstandard parameters

We present a generalization of multiple orthogonal polynomials of types I and II, which we call multiple orthogonal polynomials of mixed type. Some basic properties are formulated, and a Riemann-Hilbert problem for the multiple orthogonal polynomials of mixed type is given. We derive a Christoffel-Darboux formula for these polynomials using the solution of the Riemann-Hilbert problem. The main motivation for studying these polynomials comes from a model of non-intersecting one-dimensional Brownian motions with a given number of starting points and endpoints. The correlation kernel for the positions of the Brownian paths at any intermediate time coincides with the Christoffel-Darboux kernel for the multiple orthogonal polynomials of mixed type with respect to Gaussian weights.