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arXiv · 1512.06599

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

Jean-Paul Blaizot·Jacek Grela·Maciej A. Nowak·Wojciech Tarnowski·Piotr Warchoł
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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.

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Jean-Paul Blaizot, Jacek Grela, Maciej A. Nowak, Wojciech Tarnowski, Piotr Warchoł. 2016-01-04. Ornstein-Uhlenbeck diffusion of hermitian and non-hermitian matrices - unexpected links. https://doi.org/10.1088/1742-5468%2F2016%2F05%2F054037

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