The Cauchy two-matrix model

Abstract:

We will introduce a statistical ensemble of positive definite Hermitean random matrices whose joint probability density is coupled by a determinantal interaction. The model has properties which lie in between the Mehta-Eynard-Itzykson-Zuber two matrix model and the more common one-matrix model.

In particular we will show how the model is solvable in terms of certain biorthogonal polynomials and how the correlation functions can be computed in terms of four kernels. These latter can be computed in terms of the solution of a single Riemann--Hilbert problem. In the scaling limits, a three sheeted Riemann surface arises naturally from the solution of a vector potential problem, which proves essential in the Deift-Zhou steepest descent analysis proof of universality results.