This is a review of the Riemann-Hilbert approach to the large $N$ asymptotics in random matrix models and its applications. We discuss the following topics: random matrix models and orthogonal polynomials, the Riemann-Hilbert approach to the large $N$ asymptotics of orthogonal polynomials and its applications to the problem of universality in random matrix models, the double scaling limits, the large $N$ asymptotics of the partition function, and random matrix models with external source.
This is an introduction to the geometry of compact Riemann surfaces, largely following the books Farkas-Kra, Fay, Mumford Tata lectures. 1) Defining Riemann surfaces with atlases of charts, and as locus of solutions of algebraic equations. 2) Space of meromorphic functions and forms, we classify them with the Newton polygon. 3) Abel map, the Jacobian and Theta functions. 4) The Riemann--Roch theorem that computes the dimension of spaces of functions and forms with given orders of poles and zeros. 5) The moduli space of Riemann surfaces, with its combinatorial representation as Strebel graphs, …
This is a graduate-level introduction to C*-algebras, Hilbert C*-modules, vector bundles, and induced representations of groups and C*-algebras, with applications to quantization theory, phase space localization, and configuration space localization. The reader is supposed to know elementary functional analysis and quantum mechanics.
This paper is based on author's lectures at Kyoto University in 2010 Summer, and in the 6th MSJ-SI `Development of Moduli Theory' at RIMS in June 2013. The purpose of lectures was to review several results on Hilbert schemes of points which were obtained after author's lecture note was written. Among many results, we choose those which are about equivariant homology groups $H^T_*(X^{[n]})$ of Hilbert schemes of points on the affine plane $X = \mathbb C^2$ with respect to the torus action.
These lectures introduce the method of nonlinear steepest descent for Riemann-Hilbert problems. This method finds use in studying asymptotics associated to a variety of special functions such as the Painlevé 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. These lectures highlight a few such applications.
The Cauchy problem for the Camassa -- Holm equation with step-like initial conditions is reformulated as a Riemann -- Hilbert problem. The the initial value problem solution is obtained then in a parametric form from the Riemann -- Hilbert problem solution.
Solutions to the Riemann-Hilbert problems with irregular singularities naturally associated to semisimple Frobenius manifold structures on Hurwitz spaces (moduli spaces of meromorphic functions on Riemann surfaces) are constructed. The solutions are given in terms of meromorphic bidifferentials defined on the underlying Riemann surface. The relationship between different classes of Frobenius manifold structures on Hurwitz spaces (real doubles, deformations) is described on the level of the corresponding Riemann-Hilbert problems.
We show, how the Riemann-Hilbert approach to the elastodynamic equations, which have been suggested in our preceding papers, works in the half-plane case. We pay a special attention to the appearance of the Rayleigh waves within the scheme.
We revisit the Helmholts equation in a quarter-plane in the framework of the Riemann-Hilbert approach to linear boundary value problems suggested in late 90s by A. Fokas. We show the role of the Sommerfeld radiation condition in Fokas's scheme.
In this paper we study the Fuchsian Riemann-Hilbert (inverse monodromy) problem corresponding to Frobenius structures on Hurwitz spaces. We find a solution to this Riemann-Hilbert problem in terms of integrals of certain meromorphic differentials over a basis of an appropriate relative homology space, study the corresponding monodromy group and compute the monodromy matrices explicitly for various special cases.
This paper aims to present an application of Riemann-Hilbert approach to treat higher-order nonlinear differential equation that is an eighth-order nonlinear Schrodinger equation arising in an optical fiber. Starting from the spectral analysis of the Lax pair, a Riemann-Hilbert problem is formulated. Then by solving the obtained Riemann-Hilbert problem under the reflectionless case, N-soliton solution is generated for the eighth-order nonlinear Schrodinger equation. Finally, the three-dimensional plots and two-dimensional curves with specific choices of the involved parameters are made to show…
Under investigation in this work is the coupled Hirota system arising in nonlinear fiber. The spectral analysis of the Lax pair is first carried out and a Riemann-Hilbert problem is described. Then in the framework of the obtained Riemann-Hilbert problem with the reflectionless case, N-soliton solution is presented for the coupled Hirota system. Finally, via proper choices of the involved parameters, a few plots are made to exhibit the localized structures and dynamic characteristics of one-soliton solution.
A fifth-order nonlinear Schrodinger equation which describes one-dimensional anisotropic Heisenberg ferromagnetic spin chain is under exploration in this paper. Starting from the spectral analysis of the Lax pair, a Riemann-Hilbert problem is set up. After solving the obtained Riemann-Hilbert problem with reflectionless case, we systematically derive multisoliton solutions for the fifth-order nonlinear Schrodinger equation. In addition, the localized structures and dynamic behaviors of one- and two-soliton solutions are shown vividly via a few plots.
Two-phase solutions of focusing NLS equation are classically constructed out of an appropriate Riemann surface of genus two, and expressed in terms of the corresponding theta-function. We show here that in a certain limiting regime such solutions reduce to some elementary ones called "Solitons on unstable condensate". This degeneration turns out to be conveniently studied by means of basic tools from the theory of Riemann-Hilbert problems. In particular no acquaintance with Riemann surfaces and theta-function is required for such analysis.
In these lectures three different methods of computing the asymptotic expansion of a Hermitian matrix integral is presented. The first one is a combinatorial method using Feynman diagrams. This leads us to the generating function of the reciprocal of the order of the automorphism group of a tiling of a Riemann surface. The second method is based on the classical analysis of orthogonal polynomials. A rigorous asymptotic method is established, and a special case of the matrix integral is computed in terms of the Riemann $ζ$-function. The third method is derived from a formula for the $τ$-functio…
We find a local $(d+1) \times (d+1)$ Riemann-Hilbert problem characterizing the skew-orthogonal polynomials associated to the partition function of the Gaussian Orthogonal Ensemble of random matrices with a potential function of degree $d$. Our Riemann-Hilbert problem is similar to a local $d \times d$ Riemann-Hilbert problem found by Kuijlaars and McLaughlin characterizing the bi-orthogonal polynomials. This gives more motivation for finding methods to compute asymptotics of high order Riemann-Hilbert problems, and brings us closer to finding asymptotics of the skew-orthogonal polynomials.
In the application of the Deift-Zhou steepest descent method to the Riemann-Hilbert problem for orthogonal polynomials, a model Riemann-Hilbert problem that appears in the multi-cut case is solved with the use of hyperelliptic theta functions. We present here an alternative approach which uses meromorphic differentials instead of theta functions to construct the solution of the model Riemann-Hilbert problem.
We characterize the biorthogonal polynomials that appear in the theory of coupled random matrices via a Riemann-Hilbert problem. Our Riemann-Hilbert problem is different from the ones that were proposed recently by Ercolani and McLaughlin, Kapaev, and Bertola et al. We believe that our formulation may be tractable to asymptotic analysis.
This paper focuses on investigation of the N-coupled Hirota equations arising in an optical fiber. Starting from analyzing the spectral problem, a kind of matrix Riemann-Hilbert problem is formulated strictly on the real axis. Then based on the resulting matrix Riemann-Hilbert problem under the constraint of no reflection, multi-soliton solutions to the N-coupled Hirota equations are presented explicitly.
In this work, the inverse scattering transform of the three-component coupled Sasa-Satsuma equation is investigated via the Riemann-Hilbert method. Firstly we consider a Lax pair associated with a $7\times 7$ matrix spectral problem for the equation. Then we present the spectral analysis of the Lax pair, from which a kind of Riemann-Hilbert problem is formulated. Moreover, $N$-soliton solutions to the equation are constructed through a particular Riemann-Hilbert problem with vanishing scattering coefficients. Finally, the dynamics of the soliton solutions are discussed with some graphics.
By the classical genus zero Sugawara construction one obtains from admissible representations of affine Lie algebras (Kac-Moody algebras of affine type) representations of the Virasoro algebra. In this lecture first the classical construction is recalled. Then, after giving a review on the global multi-point algebras of Krichever-Novikov type for compact Riemann surfaces of arbitrary genus, the higher genus Sugawara construction is introduced. Finally, the lecture reports on results obtained in joint work with O.K. Sheinman. We were able to show that also in the higher genus, multi-point situa…
We prove a nonlinear steepest descent theorem for Riemann-Hilbert problems with Carleson jump contours and jump matrices of low regularity and slow decay. We illustrate the theorem by deriving the long-time asymptotics for the mKdV equation in the similarity sector for initial data with limited decay and regularity.
Explicit solutions to the Riemann-Hilbert problem will be found realising some irreducible non-rigid local systems. The relation to isomonodromy and the sixth Painleve equation will be described. Keywords: Riemann-Hilbert problem, Painleve equations, algebraic solutions, Heun equations, tetrahedral/octahedral group, triangle groups, Belyi maps.
The geometric phase that appears in the effects of Aharonov-Bohm type is interpreted in the frame of Deligne's version of the Riemann-Hilbert correspondence. We extend also the concept of flat gauge field to $B$-branes which are coherent sheaves, so that such a field on a sheaf turns it into a holonomic regular $D$-module.
In this paper, we study the theory of orthogonal trigonometric polynomials (OTP). We obtain asymptotics of OTP with positive and analytic weight functions by Riemann-Hilbert approach and find they have relations with orthogonal polynomials on the unit circle (OPUC). By the relations and the theory of OPUC, we also get four-terms recurrent formulae, Christoffel-Darboux formula and some properties of zeros for orthogonal trigonometric polynomials.
We describe the close connection between the linear system for the sixth Painlevé equation and the general Heun equation, formulate the Riemann-Hilbert problem for the Heun functions and show how, in the case of reducible monodromy, the Riemann-Hilbert formalism can be used to construct explicit polynomial solutions of the Heun equation.
A Riemann-Hilbert problem for a $q$-difference Painlevé equation, known as $q\textrm{P}_{\textrm{IV}}$, is shown to be solvable. This yields a bijective correspondence between the transcendental solutions of $q\textrm{P}_{\textrm{IV}}$ and corresponding data on an associated $q$-monodromy surface. We also construct the moduli space of $q\textrm{P}_{\textrm{IV}}$ explicitly.
We consider the gap probability for the Bessel process in the single-time and multi-time case. We prove that the scalar and matrix Fredholm determinants of such process can be expressed in terms of determinants of integrable kernels à la Its-Izergin-Korepin-Slavnov and thus related to suitable Riemann-Hilbert problems. In the single-time case, we construct a Lax pair formalism and we derive a Painlevé III equation related to the Fredholm determinant.
In this paper, we are going to solve nonlinear nonlocal reverse-time six-component six-order AKNS system. We used reverse-time reduction to reduce the coupled system to an integrable six-order NLS-type equation. Starting from the spectral problem of the AKNS system, a Riemann-Hilbert problem will be formulated. This formulation allows to generate soliton solutions by using the vectors lying in the kernel of the matrix Jost solutions. When reflection coefficients are zeros, the jump matrix is identity and the corresponding Riemann-Hilbert problem yields soliton solutions, leading to explore the…
We consider the gap probability for the Generalized Bessel process in the single-time and multi-time case. We prove that the scalar and matrix Fredholm determinants of such process can be expressed in terms of determinants of Its-Izergin-Korepin-Slavnov integrable kernels and thus related to suitable Riemann-Hilbert problems. In the single-time case, we construct a Lax pair formalism, while in the multi-time case we explicitly define a new multi-time kernel to study.
In this paper, we aim to investigate the mixed Chen-Lee-Liu derivative nonlinear Schrödinger(CLL-NLS) equation via the Riemann-Hilbert(RH) method. we construct a RH problem base on the Jost solution of the Lax pair. By solving this RH problem corresponding to the non reflection case, the N-soliton solution of CLL-NLS equation is obtained, which expression is the ratio of $(2N+1)\times(2N+1)$ determinant and $2N\times2N$ determinant.
In this paper we solve an arbitrary matrix Riemann-Hilbert (inverse monodromy) problem with quasi-permutation monodromy representations outside of a divisor in the space of monodromy data. This divisor is characterized in terms of the theta-divisor on the Jacobi manifold of an auxiliary compact Riemann surface realized as an appropriate branched covering of $\CP1$ . The solution is given in terms of a generalization of Szegö kernel on the Riemann surface. In particular, our construction provides a new class of solutions of the Schlesinger system. The isomonodromy tau-function of these solution…
We consider matrix functions with certain invariance under inversion in the unit circle. If such a function satisfies a positivity assumption on the unit circle, then only zero partial indices appear in its Riemann-Hilbert (Wiener-Hopf) factorization. It implies the unique solvability of a certain class of Riemann-Hilbert boundary value problems. It includes the ones associated with the inverse scattering transform of the focusing/defocusing integrable discrete nonlinear Schrödinger equations.
We prove a Tracy-Widom type formula for the generating function of occupancy numbers on several disjoint intervals of the higher order Airy point processes. The formula is related to a new vector-valued Painlevé II hierarchy we define, together with its Lax pair.
A GBDT version of the Backlund-Darboux transformation is constructed for a non-isospectral canonical system, which plays essential role in the theory of random matrix models. The corresponding Riemann-Hilbert problem is treated and some explicit formulas are obtained. A related inverse problem is formulated and solved.
In this work, we investigated a combined Chen-Lee-Liu derivative nonlinear Schrödinger equation(called CLL-NLS equation by Kundu) on the half-line by unified transformation approach. We gives spectral analysis of the Lax pair for CLL-NLS equation, and establish a matrix Riemann-Hilbert problem, so as to reconstruct the solution $r(z,t)$ of the CLL-NLS equation by solving. Furthermore, the spectral functions are not independent, but enjoy by a compatibility condition, which is the so-called global relation.
The purpose of this paper is to push forward the theory of operator-valued Riemann Hilbert problems and demonstrate their effectiveness in respect to the implementation of a non-linear steepest descent method \textit{á la} Deift-Zhou. In the present paper, we demonstrate that the operator-valued Riemann--Hilbert problem arising in the characterisation of so-called $c$-shifted integrable integral operators allows one to extract the large-$x$ asymptotics of the Fredholm determinant associated with such operators.
We show how the Riemann-Hilbert problem can be used to compute correlation kernels for determinantal point processes arising in different models of asymptotic combinatorics and representation theory. The Whittaker kernel and the discrete Bessel kernel are computed as examples.
A new concept of meromorphic $Σ$-factorization, for Hölder continuous functions defined on a contour $Γ$ that is the pullback of $\dot{\mathbb{R}}$ (or the unit circle) in a Riemann surface $Σ$ of genus 1, is introduced and studied, and its relations with holomorphic $Σ$-factorization are discussed. It is applied to study and solve some scalar Riemann-Hilbert problems in $Σ$ and vectorial Riemann-Hilbert problems in $\mathbb{C}$, including Wiener-Hopf matrix factorization, as well as to study some properties of a class of Toeplitz operators with $2 \times 2$ matrix symbols.
We prove the existence of the double scaling limit in the unitary matrix model with quartic interaction, and we show that the correlation functions in the double scaling limit are expressed in terms of the integrable kernel determined by the psi-function for the Hastings-McLeod solution to the Painlevé II equation. The proof is based on the Riemann-Hilbert approach.
A new four-component nonlinear Schrödinger equation is first proposed in this work and studied by Riemann-Hilbert approach. Firstly, we derive a Lax pair associated with a $5\times5$ matrix spectral problem for the four-component nonlinear Schrödinger equation. Then based on the Lax pair, we analyze the spectral problem and the analytical properties of the Jost functions, from which the Riemann-Hilbert problem of the equation is successfully established. Moreover, we obtain the $N$-soliton solutions of the equation by solving the Riemann-Hilbert problem without reflection. Finally, we derive t…
We use discrete analogs of Riemann-Hilbert problem's methods to derive the discrete Bessel kernel which describes the poissonized Plancherel measures for symmetric groups. To do this we define discrete analogs of a Riemann-Hilbert problem and of an integrable integral operator and show that computing the resolvent of a discrete integrable operator can be reduced to solving a corresponding discrete Riemann-Hilbert problem. We also give an example, explicitly solvable in terms of classical special functions, when a discrete Riemann-Hilbert problem converges in a certain scaling limit to a conven…
We present an inverse scattering transform approach to the Cauchy problem on the line for the Degasperis--Procesi equation $u_t-u_{txx}+3ωu_x+4uu_x=3u_xu_{xx}+uu_{xxx}$ in the form of an associated Riemann-Hilbert problem. This approach allows us to give a representation of the solution to the Cauchy problem, which can be efficiently used in studying its long-time behavior.
We present an inverse scattering transform approach for the equation $u_{txx}-3u_x+3u_xu_{xx}+uu_{xxx}=0$. This equation can be viewed as the short wave model for the Degasperis-Procesi equation or the differentiated Ostrovsky-Vakhnenko equation. The approach is based on an associated Riemann-Hilbert problem, which allows us to give a representation for the classical (smooth) solution, to get the principal term of its long time asymptotics, and also to describe loop soliton solutions.
We study the asymptotic behavior of oscillatory Riemann-Hilbert problems arising in the AKNS hierarchy of integrable nonlinear PDE's. Our method is based on the Deift-Zhou nonlinear steepest descent method in which the given Riemann-Hilbert problem localizes to small neighborhoods of stationary phase points. In their original work, Deift and Zhou only considered analytic phase functions. Subsequently Varzugin extended the Deift-Zhou method to a certain restricted class of non-analytic phase functions. In this paper, we extend Varzugin's method to a substantially more general class of non-analy…
In this article we calculate the dimension of the Hilbert space of Kahler quantization of the moduli space of vortices on a Riemann surface. This dimension is given by the holomorphic Euler characteristic of the quantum line bundle.
We consider the local field dynamical temperature correlation function of the Quantum Nonlinear Schrodinger equation with the finite coupling constant. This correlation function admits a Fredholm determinant representation. The related operator-valued Riemann--Hilbert problem is used for analysing the leading term of the large time and long distance asymptotics of the correlation function.
