The purpose of this paper is to give a proof to the equivalence of the derived category of holonomic systems and that of constructible sheaves. Let X be a paracompact complex manifold and let ® x and 0 x be the sheaf of differential operators and holomorphic functions, respectively. We denote by Mod(^z) the abelian category of left ^^-Modules and by D(^) its derived category. Let ~D^(^x) denote...

We consider n non-intersecting Brownian motions with two fixed starting positions and two fixed ending positions in the large n limit. We show that in case of ‘large separation’ between the endpoints, the particles are asymptotically distributed in two separate groups, with no interaction between them, as one would intuitively expect. We give a rigorous proof using the Riemann-Hilbert formalism...

It has been shown by Strahov and Fyodorov that averages of products and ratios of characteristic polynomials corresponding to Hermitian matrices of a unitary ensemble, involve kernels related to orthogonal polynomials and their Cauchy transforms. We will show that, for the unitary ensemble 1 Ẑn | detM |2αe−nV dM of n×n Hermitian matrices, these kernels have universal behavior at the origin of t...

Recent advances in the numerical solution of Riemann–Hilbert problems allow for the implementation of a Cauchy initial value problem solver for the Korteweg–de Vries equation (KdV) and the defocusing modified Korteweg–de Vries equation (mKdV), without any boundary approximation. Borrowing ideas from the method of nonlinear steepest descent, this method is demonstrated to be asymptotically accur...

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 lar...

We analyse an initial-boundary value problem for the mKdV equation on a finite interval by expressing the solution in terms of the solution of an associated matrix Riemann-Hilbert problem in the complex k-plane. This Riemann-Hilbert problem has explicit (x, t)-dependence and it involves certain functions of k referred to as “spectral functions”. Some of these functions are defined in terms of t...

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 lar...