Extreme (eigen)values under global constraint: truncations of Haar unitary matrices and Gaussian analytic functions
A talk in the Seminar Zufallsmatrizen series by
Boris Khoruzhenko
| Abstract: | Let $c_k$ be independent standard complex normals. This talk is about zeros of Gaussian Analytic Functions $f(z) = \sum c_k z^k$, conditioned by the event that $f(0)=a$. The probability law of the zero set of $f(z)$ can be derived from that of spectral outliers of random sub-unitary matrices. I will explain how this link can be used to obtain the full conditional distribution of radial zero counting function, prove its asymptotic normality and develop precise asymptotic form of the probabilities of moderate to large deviations from normality. One also finds tail probabilities for the conditional distribution of the k-th smallest absolute value of zeros for which the probability model is one of the order statistics of infinite sequence of independent random variables satisfying a global constraint. This model exhibits interesting features. For instance, the asymptotic expansion for the conditional hole probability has two built-in two scales (slow and fast) and, to leading order the conditional hole probability coincides with that of the unconditioned GAF of the form $\sum \sqrt{k+1} c_k z^k$. My talk is based on joint work with Yan Fyodorov and Thomas Prellberg (arXiv:2412.06086 [1]). Within the CRC this talk is associated to the project(s): C6 |