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).