Non-selfadjoint random matrices and randomly coupled differential equations
Spectral statistics of random matrices often exhibit universal behaviour as the dimension grows to infinity. On the global scale of the entire spectrum the empirical eigenvalue distribution concentrates around a deterministic limit; while on the smallest local scale of the eigenvalue spacing, the k-point correlation functions become universal, depending only on the symmetry class of the matrix, but not on any model details. For selfadjoint models this fact has been established in great generality. However, for non-selfadjoint models with eigenvalues in the complex plane their inherent spectral instability poses a major challenge for the study of local eigenvalue statistics. We present recent results on eigenvalue spectra for non-selfadjoint random matrices down to all scales above the eigenvalue spacing and their application to systems of randomly coupled differential equations that are used to model a wide range of disordered dynamical systems ranging from neural networks to food webs.
[Joint work with the Johannes Alt, László Erdos and David Renfrew]
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