title: Dysonian dynamics of the Ginibre ensemble.
abstract: We study 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 nontrivial way, leading to a system of coupled nonlinear equations resembling those for turbulent systems. We formulate a mathematical framework allowing simultaneous description of the flow of eigenvalues and eigenvectors, and we unravel a hidden dynamics as a function of a new complex variable, which in the standard description is treated as a regulator only. We solve the evolution equations for large matrices and demonstrate that the nonanalytic behavior of the Green's functions is associated with a shock wave stemming from a Burgers-like equation describing correlations of eigenvectors. We conjecture that the hidden dynamics that we observe for the Ginibre ensemble is a general feature of non-Hermitian random matrix models and is relevant to related physical applications.
authors.affiliation: M. Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Centre, Jagiellonian University, PL-30-059 Cracow, Poland.
authors.forename: Zdzislaw
authors.initials: Z
authors.lastname: Burda
authors.affiliation: M. Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Centre, Jagiellonian University, PL-30-059 Cracow, Poland.
authors.forename: Jacek
authors.initials: J
authors.lastname: Grela
authors.affiliation: M. Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Centre, Jagiellonian University, PL-30-059 Cracow, Poland.
authors.forename: Maciej A
authors.initials: MA
authors.lastname: Nowak
authors.affiliation: M. Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Centre, Jagiellonian University, PL-30-059 Cracow, Poland.
authors.forename: Wojciech
authors.initials: W
authors.lastname: Tarnowski
authors.affiliation: M. Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Centre, Jagiellonian University, PL-30-059 Cracow, Poland.
authors.forename: Piotr
authors.initials: P
authors.lastname: Warchoł