Numerical solution of Riemann-Hilbert problems: random matrix theory and orthogonal polynomials (Q485294)

The authors develop numerical methods for solving Riemann-Hilbert problems which are applied to the calculation of orthogonal polynomials. Spectral densities and gap statistics for a large family of finite-dimensional unitary invariant ensembles are calculated by applying the suggested approach. The considered techniques are related to the method of nonlinear steepest descent, including contour deformations and the g-function technique. The accuracy of the algorithm is investigated in detail and the results are illustrated in several examples, for example, the Hastings-McLeod solution of the homogeneous Painlevé II equation is computed.