Unveiling the significance of eigenvectors in diffusing nonHermitian matrices by identifying the underlying Burgers dynamics
Abstract
Following our recent letter [1], we study in detail an entrywise diffusion of nonhermitian complex matrices. We obtain an exact partial differential equation (valid for any matrix size N and arbitrary initial conditions) for evolution of the averaged extended characteristic polynomial. The logarithm of this polynomial has an interpretation of a potential which generates a Burgers dynamics in quaternionic space. The dynamics of the ensemble in the large N limit is completely determined by the coevolution of the spectral density and a certain eigenvector correlation function. This coevolution is best visible in an electrostatic potential of a quaternionic argument built of two complex variables, the first of which governs standard spectral properties while the second unravels the hidden dynamics of eigenvector correlation function. We obtain general formulas for the spectral density and the eigenvector correlation function for large N and for any initial conditions. We exemplify our studies by solving three examples, and we verify the analytic form of our solutions with numerical simulations.
 Publication:

Nuclear Physics B
 Pub Date:
 August 2015
 DOI:
 10.1016/j.nuclphysb.2015.06.002
 arXiv:
 arXiv:1503.06846
 Bibcode:
 2015NuPhB.897..421B
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 24 pages, 11 figures