Full Dysonian dynamics of the complex Ginibre ensemble

Abstract

We find stochastic equations governing eigenvalues and eigenvectors of a dynamical complex Ginibre ensemble reaffirming the intertwined role played between both sets of matrix degrees of freedom. We solve the accompanying Smoluchowski–Fokker–Planck equation valid for any initial matrix. We derive evolution equations for the averaged extended characteristic polynomial and for a class of k-point eigenvalue correlation functions. From the latter we obtain a novel formula for the eigenvector correlation function which we inspect for Ginibre and spiric initial conditions and obtain macro- and microscopic limiting laws.