Gernot Akemann

Planar orthogonal polynomials and random matrices: Examples for local and global universality

The complex eigenvalues of non-Hermitian random matrices typically form determinantal point processes in the complex plane. The corresponding kernel in turn relates to orthogonal polynomials in the complex plane. A prime example is the elliptic Ginibre ensemble with complex normal matrix elements, leading to planar Hermite polynomials. Two further examples will be discussed, planar Laguerre polynomials which are known to results from a chiral random matrix ensemble, and planar Gegenbauer polynomials, for which we do not have a random matrix representation yet. This detailed knowledge and the possibility to interpolate between complex and real eigenvalues allows us to take various limits. Old and new universality classes at weak non-Hermiticity, a non-Hermitian generalisation of the Marchenko-Pastur distribution and local statistics at a multi-critical point in the complex plane are amongst them.

This is joint work with Sung-Soo Byun and Nam-Gyu Kang as well as with Taro Nagao, Mario Kieburg, Ivan Parra and Graziano Vernizzi.