Ornstein-Uhlenbeck diffusion of hermitian and non-hermitian matrices - unexpected links

TL;DR

This paper compares the Ornstein-Uhlenbeck processes for hermitian and non-hermitian matrices, revealing new insights into their eigenvalue and eigenvector dynamics using generalized Green's functions.

Contribution

It introduces a novel framework involving a hidden complex variable to analyze the evolution of non-hermitian systems and uncovers unexpected links between hermitian and non-hermitian ensembles.

Findings

Coupling between eigenvalue flow and eigenvector flow in non-hermitian systems

Identification of a new complex variable crucial for understanding non-hermitian dynamics

Discovery of unexpected links between hermitian and non-hermitian ensembles

Abstract

We compare the Ornstein-Uhlenbeck process for the Gaussian Unitary Ensemble to its non-hermitian counterpart - for the complex Ginibre ensemble. We exploit the mathematical framework based on the generalized Green's functions, which involves a new, hidden complex variable, in comparison to the standard treatment of the resolvents. This new variable turns out to be crucial to understand the pattern of the evolution of non-hermitian systems. The new feature is the emergence of the coupling between the flow of eigenvalues and that of left/right eigenvectors. We analyze local and global equilibria for both systems. Finally, we highlight some unexpected links between both ensembles.