Kac-Rice inspired approach to non-Hermitian random matricesOberseminar Stochastics

by Prof. Yan Fyodorov (King’s College London)

Thursday Jul 10, 2025, 4:30 PM → 5:30 PM Europe/Berlin

Endenicher Allee 60/1-016 - Lipschitzsaal (Mathezentrum)

Beyond the framework of standard Ginibre ensembles one of the general tools
available for studying non-Hermitian random matrices is the "Hermitization
Trick" due to Girko. I will describe an alternative approach based on Kac-Rice
counting formulas which provides access not only to eigenvalues but also to
nontrivial right eigenvectors of non-Hermitian random matrices. To illustrate
power of the approach I will consider a family of matrices interpolating be-
tween complex Ginibre and real Ginibre ensembles, which in particular allows
one to reveal a new scaling regime of "weak non-reality" arising as $N → \infty$.
This part will be based on the recent preprint arXiv:2506.21058. If time allows,
I will also briefly discuss some results on notoriously difficult case of complex
symmetric matrices, obtained by the same method in collaboration with Ger-
not Akemann and Dmitry Savin.