The riemannhilbert problem, orthogonal polynomials, and random matrix theory samuel bieri i give a short introduction to the riemannhilbert problem and its connection to orthogonal polynomials and the theory of random matrices. Hilbert problem is uniquely solved in terms of the matrix szego polynomials and its cauchy transforms. Our riemannhilbert problem is different from the ones that were proposed recently by. Semiclassical asymptotics of orthogonal polynomials, riemann.
Pdf riemannhilbert problem for biorthogonal polynomials. Pdf the riemannhilbert problem, orthogonal polynomials, and. In this section, we will apply the deiftzhou steepest decent method to approximate this riemannhilbert problem by a riemannhilbert problem that is solvable explicitly. Riemannhilbert characterization for matrix orthogonal polynomials.
This is an introduction to the asymptotic analysis of orthogonal polynomials based on the steepest descent method for riemannhilbert problems. The riemann hilbert problem was obtained earlier in 24 from a reformulation of the biorthogonality 5. Riemannhilbert problems, orthogonal polynomials and computation. Generalized orthogonal polynomials, discrete kp and. The riemannhilbert method reduces a particular problem to the reconstruction of an analytic function from jump conditions or, equiv. A riemannhilbert approach to the akhiezer polynomials. This is the case where w is a polynomial of degree 4. May 15, 2008 read a riemannhilbert problem for skeworthogonal polynomials, journal of computational and applied mathematics on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. The problem that an analytic function could be completely. A riemannhilbert approach to asymptotic questions for. Random matrices, multiorthogonal polynomials and riemann. We show how the fokasitskitaev riemann hilbert problem for orthogonal polynomials with exponentially varying weights can be deformed by systematic steps reminiscent of the method of steepest descent for contour integrals.
Generalized orthogonal polynomials, discrete kp and riemann. This gives more motivation for finding methods to compute asymptotics of high order riemannhilbert problems, and brings us closer to finding full asymptotic expansions of the. The deformed riemannhilbert problem is simple enough that its structure suggests an approximate solution. Nauk sssr 266 788791 mathscinet vladimirov v s and iv volovich 1982 soviet math.
The number of solutions or solvability conditions depends on the index, and in certain singular. Riemannhilbert techniques in the theory of orthogonal. The fokasitskitaev riemann hilbert problem has wellunderstood applications to the asymptotic analysis of both orthogonal polynomials and random matrices. Global fluctuations for multiple orthogonal polynomial ensembles. Rh problem for multiple orthogonal polynomials, is to try and reduce it to a lower dimensional problem. A general framework for solving riemannhilbert problems. I was very excited with the evolution of the riemannhilbert approach to the asymptotic analysis of orthogonal polynomials, and discussed extensively with ed the new perspectives. In the nal part we use the steepest descent method in the asymptotic analysis of. The deformed riemann hilbert problem is simple enough that its structure suggests an approximate solution.
In general, the matrix riemannhilbert problem, which from now on will be referred to asrhproblem,isde. In the early nineties, fokas, its and kitaev observed that there is a natural riemannhilbert problem for 2 x. Vladimirov v s and iv volovich 1982 the wienerhopf equation, the riemann hilbert problem and orthogonal polynomials dokl. A riemannhilbert problem for skeworthogonal polynomials a riemannhilbert problem for skeworthogonal polynomials pierce, virgil u. This includes the result that the numerical approximation is uniformly accurate when the contours are. A general framework for solving riemannhilbert problems numerically sheehan olver oxford university mathematical institute 2429 st giles oxford, uk sheehan. Pdf riemannhilbert problems for multiple orthogonal. This method finds use in studying asymptotics associated to a variety of special functions such as the painleve equations and orthogonal polynomials, in solving the inverse scattering problem for certain integrable systems, and in proving universality for certain classes of random matrix ensembles. This volume expands on a set of lectures held at the courant institute on riemannhilbert problems, orthogonal polynomials, and random matrix theory. Combining this numerical algorithm with the approach of bornemann to compute fredholm determinants, we are able to calculate spectral densities and gap statistics for a broad class of. The riemannhilbert analysis to the pollaczekjacobi type. In x3 a discrete matrix equation, for which the recursion coe cients of the freud mops are solutions, is derived and it is also proven that its singularities are con ned.
Pdf i give a short introduction to the riemann hilbert problem and its connection to orthogonal polynomials and the theory of random matrices. Varvaruca, a complex analysis approach to the problem of steady gravity. Pdf i give a short introduction to the riemannhilbert problem and its connection to orthogonal polynomials and the theory of random. Riemannhilbert problem encyclopedia of mathematics. Combining this numerical algorithm with the approach of bornemann to compute fredholm determinants, we are able to calculate spectral densities and gap statistics for a broad class. This riemannhilbert problem was later used by deift et al.
The riemannhilbert problemfor orthogonal matrix polynomials scalar and additive rhp n 1 let. The riemann hilbert problem, orthogonal polynomials, and random matrix theory samuel bieri i give a short introduction to the riemann hilbert problem and its connection to orthogonal polynomials and the theory of random matrices. Consider a riemannhilbert problem for a matrixvalued analytic function rz, n that satisfies a jump condition on the contour. In this note we derive a different riemannhilbert problem. The main ones are in the theory of singular integral equations. Riemannhilbert problem for biorthogonal polynomials article pdf available in journal of physics a general physics 3616. Riemannhilbert problems and their applications university of. Riemannhilbert problems, matrix orthogonal polynomials and. The riemannhilbert problem for multiple orthogonal polynomials was formulated in 37, and the authors gave the.
The proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemann hilbert problem. The solution of the riemannhilbert problem on a riemann surface, and the equivalent problem on the fundamental domain of an automorphic function belonging to a group of permutations, has been investigated for automorphic functions of this class. A riemannhilbert problem for biorthogonal polynomials. Jon keating, university of bristol extreme values of the riemann zetafunction on the critical line, freezing in disor. Riemannhilbert problems for multiple orthogonal polynomials. Our riemannhilbert problem is similar to a local d. Generalized orthogonal polynomials, discrete kp and riemannhilbert problems.
Riemannhilbert problems and their applications university. Pdf a riemannhilbert problem for biorthogonal polynomials. Why do very general ensembles of random n times n matrices exhibit. Hilbert problem is given as an appropriate framework for the discussion of the szego matrix and the associated szego recursion relations for the matrix orthogonal polynomials and its cauchy transforms. The fokasitskitaev riemannhilbert problem has wellunderstood applications to the asymptotic analysis of both orthogonal polynomials and random matrices. The riemannhilbert problem and inverse scattering siam. To study the asymptotic behaviors of orthogonal polynomials and the hankel determinant associated with the pollaczekjacobi type weight, we introduce a model riemannhilbert problem rhp for short for.
We employ the steepestdescenttype method introduced in 18 and further developed in 17, 19 in. It is based on a characteriz ation of orthogonal polynomials due to fokas, its, and. As an application of the semiclassical asymptotics of the orthogonal polynomials, we prove the universality of the local distribution of eigenvalues in the matrix model. Riemann hilbert problem techniques are an e cient and powerfull tool for random matrix theory which we discuss in more detail.
To relate the dual formulations of the riemannhilbert problem characterizing biorthogonal polynomials obtained by the different members of this group 1, 2, 3, 9. The orthogonal polynomial problem is formulated as a riemann hilbert problem following 22, 23. Riemannhilbert techniques in the theory of orthogonal matrix. The proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem.
The riemannhilbert problem, orthogonal polynomials, and. Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Our riemannhilbert problem is based on the fact that the biorthogonal polynomials can be characterized as multiple orthogonal polynomials see below. Shargorodsky, on a nonlinear riemannhilbert problem arising in the theory of stokes waves 3. The riemannhilbert approach to orthogonal polynomials and matrix models was suggested in 1991 by fokas, kitaev, and this author, and recently it helped in solving some of the longstanding problems in the asymptotics of orthogonal polynomials related to universalities in random matrices. We consider the relation of the multicomponent 2d toda hierarchy with the matrix orthogonal and biorthogonal polynomials. Orthogonal polynomials 75 where the yij are analytic functions on c \ r, and solve for such matrices the following matrixvalued riemannhilbert problem. Riemannhilbert problems, matrix orthogonal polynomials. A riemannhilbert problem for the finitegenus solutions of the kdv equation and its numerical solution. Dec 11, 20 recently, a general approach to solving riemannhilbert problems numerically has been developed.
We show the riemannhilbert factorization is tantamount to the factorization of the moment matrix into the product of a lower times uppertriangular matrix. A riemannhilbert problem for skeworthogonal polynomials. The technique involves reformulating the equation as a vector functionaldifference equation of the. One of these caracterizations is given by a riemann hilbert problem. Riemannhilbert analysis for orthogonal polynomials citeseerx. The discrete equation is explicitly derived in the matrix freud case, associated with matrix quartic potentials. We illustrate our method by concentrating on the charlier polynomials. Our riemann hilbert problem is di erent from the ones that were. In this paper matrix orthogonal polynomials in the real line are described in terms of a. Recently, a general approach to solving riemannhilbert problems numerically has been developed.
This approach provides an easy derivation of discrete equations for the corresponding matrix recursion coefficients. We show the riemann hilbert factorization is tantamount to the factorization of the moment matrix into the product of a lower times uppertriangular matrix. Riemannhilbert analysis for orthogonal polynomials springerlink. The goal of the course was to prove universality for a variety of statistical quantities arising in the theory of random matrix models. Riemannhilbert problems and multiple orthogonal polynomials. Lecture notes on riemannhilbert problems and multiple. The nonhomogeneous riemannhilbert problem in a somewhat different formulation was first considered by i. In the early nineties, fokas, its and kitaev observed that there is a natural riemann hilbert problem for 2 2 matrix functions associated which a system of orthogonal polynomials. The riemannhilbert problemfor orthogonal matrix polynomials hilberts 21st problem riemann. Our proof starts from an observation of fokas et al.
The formulation of a riemannhilbert problem for multiple orthogonal polynomials is due to van assche et al. In this paper, matrix orthogonal polynomials in the real line are described in terms of a riemannhilbert problem. To study the asymptotic behaviors of orthogonal polynomials and the hankel determinant associated with the pollaczekjacobi type weight, we introduce a model riemann hilbert problem rhp for short for. A result by fokas, its, and kitaev shows that the orthogonal polynomial in definition 1. We demonstrate its e ectiveness by computing solutions to other painlev e tran. This riemann hilbert problem and the deiftzhou method of nonlinear steepest descent are the key ingredients in a proof of universality for unitary invariant random matrix ensembles. This riemannhilbert problem may be tractable, at least in the sense that the biorthogonal one is, and allow one to rigorously compute the asymptotic expansion of the skeworthogonal polynomials.
The method is called steepest descent method for riemannhilbert problems. We show how the fokasitskitaev riemannhilbert problem for orthogonal polynomials with exponentially varying weights can be deformed by systematic steps reminiscent of the method of steepest descent for contour integrals. In this paper, we develop the riemannhilbert approach to study the global asymptotics of discrete orthogonal polynomials with infinite nodes. The rh problem for op the rh problem for mop outline 1 what is a riemann hilbert problem. The riemannhilbert problem rhp determined by a pair. We analyze the riemannhilbert problems by a steepest descent type.
Hilbert problems is developed, based on a recent method for the homogeneous painlev e ii riemannhilbert problem. But the behaviour of orthogonal polynomials is in turn directly related to the. Riemannhilbert problems, orthogonal polynomials and. An analytical method for scalar secondorder functionaldifference equations with meromorphic periodic coef. From the riemannhilbert problem it follows not only the recursion relations but also, for a type of matrix freud weight with m 4, a nonlinear recursion relation 58 2. Deift, p 1999, orthogonal polynomials and random matrices. For that aim following 12 we formulate the matrix riemannhilbert problem associated with the mops. A riemann hilbert problem for biorthogonal polynomials a. The riemannhilbert problem for orthogonal matrix polynomials. Riemannhilbert problems and multiple orthogonal polynomials 1 arno kuijlaars department of mathematics. Semiclassical asymptotics of orthogonal polynomials. It is shown that, when the initial condition and the. The multigraded hankel reduction of this hierarchy is co.
Suppose that there exists a matrixvalued smooth map g. We first construct a riemannhilbert problem y associated with these polynomials and then establish some technical results to. This volume expands on a set of lectures held at the courant institute on riemann hilbert problems, orthogonal polynomials, and random matrix theory. Introduction at the paris conference of the international congress for mathematicians in 1900 and later in sorbonne. Pdf the riemannhilbert problem, orthogonal polynomials. Tr mclaughliny abstract we characterize the biorthogonal polynomials that appear in the theory of coupled random matrices via a riemann hilbert problem. These lectures introduce the method of nonlinear steepest descent for riemann hilbert problems.
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