An analytical method for scalar second-order functional-difference equations with meromorphic periodic coefficients is proposed. The technique involves reformulating the equation as a vector functional-difference equation of the first order and reducing it to a scalar Riemann–Hilbert problem for two finite segments on a hyperelliptic surface. The final step of the procedure is solution of the c...

We establish universality of local eigenvalue correlations in unitary random matrix ensembles 1 Zn | det M | 2α e −ntr V (M) dM near the origin of the spectrum. If V is even, and if the recurrence coefficients of the orthogonal polynomials associated with |x| 2α e −nV (x) have a regular limiting behavior, then it is known from work of Akemann et al., and Kanzieper and Freilikher that the local ...

An elliptic analogue of the ordinary AKNS hierarchy is constructed. This gives an example of Krichever’s construction of Lax and zero-curvature equations on an algebraic curve. In addition to the usual AKNS fields, this hierarchy contains the so called Tyurin parameters as dynamical variables. The construction of the hierarchy resembles that of the ordinary AKNS hierarchy. A half of the Tyurin ...

Abstract We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same orthogonal polynomials and finally c) those constructed from both orthogonal polynomials and the...

This paper extends the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou in a critical new way. We present, in particular, an algorithm, to obtain the support of the Riemann-Hilbert problem for leading asymptotics. Applying this extended method to small dispersion KdV (Korteweg-de Vries) equation, we (i) recover the variational formulation of P. D. Lax and C. D. ...

We establish universality of local eigenvalue correlations in unitary random matrix ensembles 1 Zn | det M | 2α e −ntr V (M) dM near the origin of the spectrum. If V is even, and if the recurrence coefficients of the orthogonal polynomials associated with |x| 2α e −nV (x) have a regular limiting behavior, then it is known from work of Akemann et al., and Kanzieper and Freilikher that the local ...

We establish universality of local eigenvalue correlations in unitary random matrix ensembles 1 Zn | det M | 2α e −ntr V (M) dM near the origin of the spectrum. If V is even, and if the recurrence coefficients of the orthogonal polynomials associated with |x| 2α e −nV (x) have a regular limiting behavior, then it is known from work of Akemann et al., and Kanzieper and Freilikher that the local ...

In this letter we describe how to compute the finite-genus solutions of the Korteweg-de Vries equation using a Riemann-Hilbert problem that is satisfied by the Baker-Akhiezer function corresponding to a Schrödinger operator with finite-gap spectrum. The recovery of the corresponding finite-genus solution is performed using the asymptotics of the Baker–Akhiezer function. This method has the bene...

We give an explicit solution of a q -Riemann–Hilbert problem that arises in the theory orthogonal polynomials, prove it is unique and deduce several properties. In particular, we describe asymptotic behaviour zeros limit as degree polynomial approaches infinity.

Solutions of the Riemann-Hilbert problem implementing the twistorial structure of the dispersionless Toda (dToda) hierarchy are obtained. Two types of string equations are considered which characterize solutions arising in hodograph sectors and integrable structures of two-dimensional quantum gravity and Laplacian growth problems.