Аннотация:
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 x N.
I will first derive the general finite N expression for the JPD of a real
eigenvalue and the associated non-orthogonality factor in the real Ginibre
ensemble, and then analyze its 'bulk' and 'edge' scaling limits. I will
also discuss ongoing work on real elliptic ensembles.
The ensuing distribution is maximally heavy-tailed, so that all integer
moments beyond normalization are divergent. A similar calculation for the
associated non-orthogonality factor in the complex Ginibre ensemble yields
a distribution with the finite first moment complementing recent studies
by P. Bourgade and G. Doubach. Its 'bulk' scaling limit yields a
distribution whose first moment reproduces the well-known result of
Chalker and Mehlig (1998), and I will provide the 'edge' scaling
distribution for this case as well.
The presentation will be mainly based on the paper:
Y.V. Fyodorov, Commun. Math. Phys. 363 (2), 579-603 (2018)
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