We present a generalization of multiple orthogonal polynomials of type I and type II, which we call multiple orthogonal polynomials of mixed type. Some basic properties are formulated, and a Riemann-Hilbert problem for the multiple orthogonal polynomials of mixed type is given. We derive a Christoffel-Darboux formula for these polynomials using the solution of the Riemann-Hilbert problem. The m...

We consider the random matrix ensemble with an external source 1 Zn e−nTr( 1 2M −AM)dM defined on n×n Hermitian matrices, where A is a diagonal matrix with only two eigenvalues ±a of equal multiplicity. For the case a > 1, we establish the universal behavior of local eigenvalue correlations in the limit n → ∞, which is known from unitarily invariant random matrix models. Thus, local eigenvalue ...

In this paper Riemann–Hilbert problem is applied to the solvability of a mixed type Monge-Ampère equation and the index formula of ordinary differential equations. Blowing up onto the torus turns mixed type equations into elliptic equations, to which R-H problem is applied.

This paper presents a new uniquely solvable boundary integral equation for computing the conformal mapping, its derivative and its inverse from bounded multiply connected regions onto the five classical canonical slit regions. The integral equation is derived by reformulating the conformal mapping as an adjoint Riemann-Hilbert problem. From the adjoint Riemann-Hilbert problem, we derive a bound...

The dispersionless Toda hierarchy turns out to lie in the heart of a recently proposed Landau-Ginzburg formulation of two-dimensional string theory at self-dual compactification radius. The dynamics of massless tachyons with discrete momenta is shown to be encoded into the structure of a special solution of this integrable hierarchy. This solution is obtained by solving a Riemann-Hilbert proble...

Abstract. We develop a new asymptotic method for the analysis of matrix Riemann-Hilbert problems. Our method is a generalization of the steepest descent method first proposed by Deift and Zhou; however our method systematically handles jump matrices that need not be analytic. The essential technique is to introduce nonanalytic extensions of certain functions appearing in the jump matrix, and to...