Hosted by: Tomomi Ishikawa

I will present a study of the time evolution of Ginibre matrices whose elements undergo Brownian motion. The non-Hermitian character of the Ginibre ensemble binds the dynamics of eigenvalues to the evolution of eigenvectors in a non-trivial way, leading to a system of coupled nonlinear equations resembling those for turbulent systems. We will formulate a mathematical framework allowing simultaneous description of the flow of eigenvalues and eigenvectors, and unravel a hidden dynamics as a function of new complex variable, which in the standard description is treated as a regulator only. We shall solve the evolution equations for large matrices and demonstrate that the non-analytic behavior of the Green's functions is associated with a shock wave stemming from a Burgers-like equation describing correlations of eigenvectors. I will start by reviewing similar notions in a simpler, Hermitian setting. Joint work with Zdzislaw Burda, Jacek Grela, Maciej A. Nowak and Wojtek Tarnowski (Phys.Rev.Lett. 113 (2014) 104102).