Antti Haimi: The Polyanalytic Ginibre Ensembles

Abstract: Ginibre ensemble is the eigenvalue process of a random $N*N$ normal matrix picked from a Gaussian probability measure. Statistical properties of this process have been well studied. It is known that the correlation functions can be expressed in terms of reproducing kernels of polynomial Bargmann-Fock spaces. We represent a family of generalizations of the Ginibre ensemble, where the correlations are given by reproducing kernels of polyanalytic polynomial spaces. The global statistical properties of the processes are the same as in the Ginibre ensemble, but local statistics look different. We suggest that the process describes higher Landau levels (Ginibre ensemble corresponding to the lowest one).