Multiple orthogonal polynomials in random matrix theory

Understand

Multiple orthogonal polynomials are a generalization of orthogonal polynomials in which the orthogonality is distributed among a number of orthogonality weights.

• They appear in random matrix theory in the form of special determinantal point processes that are called multiple orthogonal polynomial (MOP) ensembles.
• The correlation kernel in such an ensemble is expressed in terms of the solution of a Riemann-Hilbert problem, that is of size (r+1) x (r+1) in the case of r weights.
• A number of models give rise to a MOP ensemble, and we discuss recent results on models of non-intersecting Brownian motions, Hermitian random matrices with external source, and the two matrix model.