Dysonian Dynamics of the Ginibre Ensemble

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.

Figure 1

The main figure shows, for a given $|z|$, the characteristics (straight lines) and caustics (dashed lines). Inside the later a shock is developed (double vertical line). Left inset shows the solution of Eq. (13) at ($\tau =|z{|}^2$). Right inset shows the caustics mapped to the $(r=|w|,z)$ hyperplane at the same moment of time. The section $r=0$ yields the circle $|z{|}^2=\tau$, bounding the domain of eigenvalues and eigenvectors correlations for the GE.

Figure 2

The figure depicts theoretical (lines) and numerical (symbols) time dependence of the Petermann factor (rescaled by $1/N$), for different values of $|z|$. For the latter, $3\times 10^4$, $200\times 200$ matrices were used.