Large n limit of Gaussian random matrices with external source, Part III: Double scaling limit

We consider the double scaling limit in the random matrix ensemble with an external source \frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM defined on $n\times n$ Hermitian matrices, where $A$ is a diagonal matrix with two eigenvalues $\pm a$ of equal multiplicities. The value $a=1$ is critical since the eigenvalues of $M$ accumulate as $n \to \infty$ on two intervals for $a > 1$ and on one interval for $0 < a < 1$. These two cases were treated in Parts I and II, where we showed that the local eigenvalue correlations have the universal limiting behavior known from unitary random matrix ensembles. For the critical case $a=1$ new limiting behavior occurs which is described in terms of Pearcey integrals, as shown by Br\'ezin and Hikami, and Tracy and Widom. We establish this result by applying the Deift/Zhou steepest descent method to a $3 \times 3$-matrix valued Riemann-Hilbert problem which involves the construction of a local parametrix out of Pearcey integrals. We resolve the main technical issue of matching the local Pearcey parametrix with a global outside parametrix by modifying an underlying Riemann surface.Comment: 36 pages, 9 figure

Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights

We study a model of $n$ non-intersecting squared Bessel processes in the confluent case: all paths start at time $t = 0$ at the same positive value $x = a$, remain positive, and are conditioned to end at time $t = T$ at $x = 0$. In the limit $n \to \infty$, after appropriate rescaling, the paths fill out a region in the $tx$-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at $x = 0$, but at a certain critical time $t^*$ the smallest paths hit the hard edge and from then on are stuck to it. For $t \neq t^*$ we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time $t$ constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a $3 \times 3$ matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large $n$ limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest.Comment: 59 pages, 11 figure

On universality of local edge regime for the deformed Gaussian Unitary Ensemble

We consider the deformed Gaussian ensemble $H_n=H_n^{(0)}+M_n$ in which $H_n^{(0)}$ is a hermitian matrix (possibly random) and $M_n$ is the Gaussian unitary random matrix (GUE) independent of $H_n^{(0)}$. Assuming that the Normalized Counting Measure of $H_n^{(0)}$ converges weakly (in probability if random) to a non-random measure $N^{(0)}$ with a bounded support and assuming some conditions on the convergence rate, we prove universality of the local eigenvalue statistics near the edge of the limiting spectrum of $H_n$.Comment: 25 pages, 2 figure

An expansion for polynomials orthogonal over an analytic Jordan curve

We consider polynomials that are orthogonal over an analytic Jordan curve L with respect to a positive analytic weight, and show that each such polynomial of sufficiently large degree can be expanded in a series of certain integral transforms that converges uniformly in the whole complex plane. This expansion yields, in particular and simultaneously, Szego's classical strong asymptotic formula and a new integral representation for the polynomials inside L. We further exploit such a representation to derive finer asymptotic results for weights having finitely many singularities (all of algebraic type) on a thin neighborhood of the orthogonality curve. Our results are a generalization of those previously obtained in [7] for the case of L being the unit circle.Comment: 15 pages, 1 figur

Relative asymptotics for orthogonal matrix polynomials with respect to a perturbed matrix measure on the unit circle

19 pages, no figures.-- MSC2000 codes: 42C05, 47A56.MR#: MR1970413 (2004b:42058)Zbl#: Zbl 1047.42021Given a positive definite matrix measure Ω supported on the unit circle T, then main purpose of this paper is to study the asymptotic behavior of L_n(\tilde{\Omega}) L_n(\Omega) -1} and \Phi_n(z, \tilde{\Omega}) \Phi_n(z, \tilde{\Omega}) -1} where $\tilde{\Omega}(z) = \Omega(z) + M \delta ( z - w)$, $1$, M is a positive definite matrix and δ is the Dirac matrix measure. Here, Ln(·) means the leading coefficient of the orthonormal matrix polynomials Φn(z; •).Finally, we deduce the asymptotic behavior of $\Phi_n(omega, \tilde{\Omega}) \Phi_n(omega, \Omega)$ in the case when M=I.The work of the second author was supported by Dirección General de Enseñanza Superior (DGES) of Spain under grant PB96-0120-C03-01 and INTAS Project INTAS93-0219 Ext.Publicad

Differential equations for the radial limits in Z+2of the solutions of a discrete integrable system

Abstract: A limiting property of the coefficients of the nearest-neighbor recurrence coefficients for the multiple orthogonal polynomials is studying. Namely, assuming existence of the limits along rays of the lattice nearest-neighbor coefficients, we describe the limit in terms of the solution of a system of ordinary differential equations. For Angelesco systems, the result is illustrated numerically.Note: Research direction:Mathematical problems and theory of numerical method

Matrix Riemann-Hilbert Problems for Pade Approximants of Orthogonal Expansions

Abstract: Markov-type functions generated by measures given on some interval are considered. We are constructing Pade approximants of orthogonal expansions for their Fourier series expansion by orthogonal polynomials on some other interval. Besides, we are studying both types of such constructions: linear Frobenius-Pade approximants and nonlinear Fourier-Pade ones. We have obtained two main new results in this paper: complete set of orthogonality relations for Fourier-Pade approximants denominators, and also equivalent reformulation of the problems concerning Pade-Fourier approximants of orthogonal expansions in terms of matrix Riemann-Hilbert problems.Note: Research direction:Mathematical problems and theory of numerical method