Ornstein-Uhlenbeck diffusion of hermitian and non-hermitian matrices - unexpected links
arXiv:1512.06599 · doi:10.1088/1742-5468/2016/05/054037
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.
21 pages, 4 figures, submitted to J. Stat. Phys
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