A non-Hermitian generalisation of the Marchenko-Pastur distribution: from the circular law to multi-criticality

Abstract: In this talk, I will discuss complex eigenvalues of the product of two rectangular complex Ginibre matrices that are correlated through a non-Hermiticity parameter.

In the first half, I will present the limiting spectral distribution of the model, which interpolates between classical results for random matrices on the global scale, the circular law and the Marchenko-Pastur distribution. In the second half, I will explain the microscopic behaviours of the model, which includes the limiting local correlation kernel at multi-criticality, where the interior of the spectrum splits into two connected components.

The global statistics follows from the solution of certain equilibrium measure problem and concentration for the 2D Coulomb gases on Frostman’s equilibrium measure, whereas the local statistics follows from a saddle point analysis of the kernel of orthogonal Laguerre polynomials in the complex plane.

This is based on joint work with Gernot Akemann and Nam-Gyu Kang.

mathematical physicscombinatoricsprobabilityrepresentation theorystatistics theory

Audience: researchers in the topic

Junior Integrable Probability Seminar

Series comments: The Junior Integrable Probability Seminar is a series of talks taking place online and involving young researchers working in Integrable Probability and related fields. Go to sites.google.com/view/junior-ips for zoom link and password for the talk a day before the seminar.