Speaker: Yan Fyodorov
Title: On statistics of bi-orthogonal eigenvectors
in real and complex Ginibre ensembles
Abstract:
I will discuss a method of studying the joint probability density
(JPD) of an eigenvalue and the associated 'non-orthogonality overlap factor'
(also known as the 'eigenvalue condition number') of the left and right
eigenvectors for non-selfadjoint Gaussian random matrices of size $N\times N$.
I will first derive the general finite $N$ expression for the JPD of a real
eigenvalue $\lambda$ and the associated non-orthogonality factor in the real
Ginibre ensemble, and then analyze its 'bulk' and 'edge' scaling limits.
The ensuing distribution is maximally heavy-tailed, so that all integer
moments beyond normalization are divergent. A similar calculation for a
complex eigenvalue $z$ and the associated non-orthogonality factor in the
complex Ginibre ensemble yields a distribution with the finite first moment.
Its 'bulk' scaling limit yields a distribution whose first moment reproduces
the well-known result of Chalker and Mehlig \cite{ChalkerMehlig1998},
and we provide the 'edge' scaling distribution for this case as well.
The presentation is based on the paper:
Y.V. Fyodorov, Commun. Math. Phys. 363 (2), 579--603 (2018)
and complements recent studies by Bourgade and Dubach (2018).