
СЕМИНАРЫ 
Семинар Добрушинской лаборатории Высшей школы современной математики МФТИ



On statistics of biorthogonal eigenvectors in nonselfadjoint Gaussian random matrices ^{}, Yan Fyodorov^{} ^{} King's College London 

Аннотация: I will discuss a method of studying the joint probability density (JPD) of an eigenvalue and the associated 'nonorthogonality overlap factor' (also known as the 'eigenvalue condition number') of the left and right eigenvectors for nonselfadjoint 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 nonorthogonality 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 heavytailed, so that all integer moments beyond normalization are divergent. A similar calculation for the associated nonorthogonality 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 wellknown 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), 579603 (2018) 