Universality in the two matrix model

Abstract:

Eigenvalue statistics of random matrices in the two-matrix model are described by two families of biorthogonal polynomials. We give a steepest descent analysis of a 4x4 matrix-valued Riemann-Hilbert problem that characterizes one of the families of biorthogonal polynomials in a special case of a quartic potential. An important ingredient in the analysis is a vector equilibrium problem involving both an external field and an upper constraint. Its minimizer describes the limiting mean distribution of the eigenvalues of one of the matrices. This is joint work with Maurice Duits.