RHPH research project

Part of the work has been described in a survey paper [A1].

The aim of the project is to apply the steepest descent method to a number of problems arising in approximation theory and mathematical physics. Building on the existing expertise we expect to be able to obtain significant contributions in the theory of orthogonal polynomials and associated random matrix models.

We are particarly interested in applying the steepest descent method to higher order situations. The Riemann-Hilbert problems that have been fully analysed are stated for 2x2 matrix valued analytic functions. We want to extend this to 3x3 or higher order matrices. Such higher order Riemann-Hilbert problem appear in coupled random matrix models and Padé-Hermite approximation (see objectives below).

The research group NALAG (Numerical Approximation and Linear Algebra Group) of the Department of Computer Science of the Katholieke Universiteit Leuven has strong expertise in the computational aspects of Padé-Hermite approximation. Numerically stable and efficient algorithms have been developed in order to compute these approximants [N5]. The interaction of theory with computation is very valuable in order to gain insight in the sometimes complex phenomena that may occur.

Based on the knowledge on (formal) orthogonal polynomials and the corresponding moment matrices with their specific structure, a number of methods have been developed to compute all zeros of an analytic function in a certain region of the complex plane. This research has lead to the software package ZEAL [N4] and the monograph [N3]. This will also be of great value for the project.

Objectives

The objective of the project is to apply the steepest descent method for Riemann-Hilbert problems of Deift and Zhou to a number of problems that are of interest in mathematics and mathematical physics.

1. Padé-Hermite approximation.

   Padé-Hermite approximants are rational approximants that approximate a number of functions simultaneously. The rational functions have a common denominator, which is a polynomial that satisfies a number of orthogonality conditions. These orthogonality conditions allow for the formulation of a Riemann-Hilbert problem that characterizes the Padé-Hermite approximants. This was done in recent research with Jeffrey Geronimo from Georgia Tech [A4]. The matrices in the Riemann-Hilbert problem have size (n+1)x(n+1) if n is the number of functions to be approximated.

   We want to analyse this Riemann-Hilbert problem with the steepest descent method of Deift and Zhou [4]. If we are successful then it will be possible to describe the error in the approximation in very much detail. This will improve many of the results that are known presently. In order to use the steepest descent method we need a good understanding of the distribution of zeros and poles of the Padé-Hermite approximants, at least at an informal level. The distribution can be very complex. Reliable numerical computations of the zeros and poles is important.

   The research group NALAG has developed methods to compute all zeros of an analytic function in a certain region of the complex plane. This research has lead to the software package ZEAL [N4] and the monograph [N3]. This will be of great value for the project.

2. Biorthogonal polynomials and coupled random matrices.

   In the theory of random matrices one is interested in the behavior of eigenvalues of large matrices distributed according to some probability measure. For quite some time it is known that the eigenvalue distribution of a large class of Hermitian random matrices can be fully described by orthogonal polynomials. More recently it has been found that coupled random matrices are connected with biorthogonal polynomials [2], [9]. This type of polynomials are less well-known than orthogonal polynomials. A Riemann-Hilbert problem that characterizes biorthogonal polynomials has been given [3], [8], [10], but it is not clear yet how the steepest descent method can be applied to the proposed Riemann-Hilbert problems. In ongoing work with K. McLaughlin, an alternative formulation of a Riemann-Hilbert problem for biorthogonal polynomials has been found. We are optimistic that the steepest descent method can be applied to this fomulation, at least for certain important special cases. The precise treatment will not be easy, but we have certain ideas in this direction.

   We expect that the behavior of zeros of certain analytic functions will be crucial for the asymptotic analysis. The accurate computation of these zeros will be an important tool. Here we can rely on the work that has been done in the research group NALAG on the computation of zeros of analytic

3. Orthogonal polynomials and eigenvalues of random matrices.

   The connection with orthogonal polynomials and the Riemann-Hilbert method has provided a proof for the universality of spacings between consecutive eigenvalue for large classes of random matrix ensembles [5], [6]. There are some other classes for which universality is known but which have not been analysed with the Riemann-Hilbert technique. As an example, we mention the chiral ensembles that play a role in quantum chromodynamics [9]. We intend to carry out such an analysis within the project. It will be necessary to perform a special local analysis near the origin and this analysis could be similar to the one carried out in [A6]. We will obtain strong asymptotics for the associated orthogonal polynomials near the origin, which in turn would prove the universality. If it works, then we expect to be able to treat the known cases, and, moreover, to extend it to a substantial more general situation. We would also obtain new results for recurrence coefficients and other quantities related to orthogonal polynomials.

   The behavior of orthogonal polynomials near the endpoint of the support of the orthogonality measure is different from the behavior in the bulk. Its analysis requires the development of special techniques which has already been done in [A2]. There is an important variation which has not been looked at in much detail. We expect that this case is related to the case of a critical quartic potential that has been treated recently in [4]. We also propose to investigate this aspect of orthogonal polynomials and the associated random matrixfunctions [N3], [N4].

Design and methodology

The project is centered around the research group Analysis of the Department of Mathematics but there is an essential input from the research group NALAG.

In recent years a major part of the research within the group Analysis has been in the direction of Riemann-Hilbert problems. The two applicants A. Kuijlaars and W. Van Assche have established collaborations with foreign experts (K. McLaughlin from North Carolina, J. Geronimo from Georgia Tech, A. Aptekarev from Moscow, F. Wielonsky from Lille) which have led to a number of substantial results.

The research group NALAG of the Department of Computer Science is strong in the computation of Padé-Hermite approximants. The applicant M. Van Barel has also been involved in the development of a software package to calculate zeros of analytic functions. Also for the mainly theoretical work in the project, it is of great importance to have strong computational tools available. Reliable numerical work will be valuable when formulating hypothesis, for example on the possible distribution of poles of Padé-Hermite approximants.

The applicants will be actively involved in the project. Since there are many open problems which are also excellent research problems for Ph.D. work, we strongly ask for a research assistant who can work fulltime and who can be funded within the project. The assistant will be trained in both the theoretical and computational parts of the project. If succesful, this research will lead to a Ph.D. degree in science.

We intend to have regularly seminars to discuss the progress of the project and to present the obtained results. We continue to have active collaboration with foreign groups, take part in international conferences, and publish in major international journals.

Relevant and recent literature

[1] J. Baik, P. Deift and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 (1999), 1119--1178.

[2] M. Bertola, B. Eynard, and J. Harnad, Duality, biorthogonal polynomials and multi-matrix models, Comm. Math. Phys. 229 (2002), 73--120.

[3] M. Bertola, B. Eynard, and J. Harnad, Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem, preprint 2002, nlin.SI/0208002.

[4] P. Bleher and A. Its, Double scaling limit in the random matrix model: the Riemann-Hilbert approach, preprint 2002, math-ph/0201003.

[5] P. Deift and X.Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems, Asymptotics for the MKdV equation, Ann. Math. 137 (1993), 295--368.

[6] P. Deift, T. Kriecherbauer, K. McLaughlin, S.Venakides and X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), 1335--1425.

[7] P. Deift, T. Kriecherbauer, K. McLaughlin, S. Venakides and X. Zhou, Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure. Appl. Math. 52 (1999), 1491--1552.

[8] N. Ercolani and K. McLaughlin, Asymptotics and integrable structures for biorthogonal polynomials associated to a random two-matrix model, Physica D 152/153 (2001), 232--268.

[9] E. Kanzieper and V. Freilikher, Spectra of large random matrices: A method of study, in: Diffuse Waves in Complex Media, edited by J. P. Fouque, NATO ASI Series Vol. 531, (Kluwer, Dordrecht, 1999), pp. 165--211.

[10] A. Kapaev, The Riemann-Hilbert problem for the bi-orthogonal polynomials, preprint 2002, nlin.SI/0207036.

Relevant work of the research group analysis

[A1] A. Kuijlaars, Riemann-Hilbert analysis for orthogonal polynomials, Lecture notes for Summer School on Orthogonal Polynomials and Special Functions, Leuven, 2002.

[A2] A.Kuijlaars, K. McLaughlin, W. Van Assche, and M. Vanlessen, The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1], preprint 2001, math.CA/0111252.

[A3] A. Kuijlaars and M. Vanlessen, Universality for eigenvalue correlations from the modified Jacobi unitary ensemble, Internat. Math. Research Notices 2002, no. 30, (2002), 1575--1600

[A4] W. Van Assche, J. Geronimo, and A.Kuijlaars, Riemann-Hilbert problems for multiple orthogonal polynomials, pp. 23--59 in ``Nato ASI Special Functions 2000" (J. Bustoz et al, eds.), Kluwer Academic Publishers, Dordrecht 2001.

[A5] A. Kuijlaars and K. McLaughlin, Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields, Communications on Pure and Applied Mathematics 53 (2000), 736--785.

[A6] M. Vanlessen, Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, preprint 2002, math.CA/0212014.

Relevant work of the research group NALAG.

[N1] A. Bultheel and M. Van Barel, Padé techniques for model reduction in linear system theory: a survey, J. Comput. Appl. Math. 14 (1986), 401--438.

[N2] G. De Samblanx, M. Van Barel, and A. Bultheel, Duality in vector Padé-Hermite approximation problems, J. Comput. Appl. Math. 66 (1996), 153--166.

[N3] P. Kravanja and M. Van Barel, Computing the zeros of analytic functions. Lecture Notes in Mathematics, 1727. Springer-Verlag, Berlin, 2000.

[N4] P. Kravanja, M. Van Barel, O. Ragos, M.N. Vrahatis, and F.A. Zafiropoulos, ZEAL: A mathematical software package for computing zeros of analytic functions, Comput. Phys. Commun. 124 (2000), 212--232.

[N5] M. Van Barel and A. Bultheel, A look-ahead method for computing vector Padé-Hermite approximants, Constr. Approx. 11 (1995), 455--476.

[N6] M. Van Barel, V. Ptak, and Z. Vavrin, Extending the notions of companion and infinite companion to matrix polynomials, Linear Alg. Appl. 290 (1999), 61--94.