Speaker
Description
I will talk about zeros of the infinite Gaussian power series $f(z)=\sum c_k z^k$ conditioned on the event that $f(0)=a$. Forrester and Ipsen 2019 showed that if the coefficients $c_k$ are independent standard complex normals then the conditional probability law of the zero set of $f(z)$ can be obtained from that of the spectrum of random subunitary matrices. I will explain how using this connection one obtains the conditional distribution of the smallest zero of $f(z)$ in terms of $q$-series and discuss its dependence on the parameter $a$. In the realm of the extreme value theory, the modulus of the smallest zero $r_{min}$ is realised as the smallest value of a sequence of independent random variables subject to a constraint. Although the conditional probability distribution of $r_{min}$ is not one of the three canonical forms, it interpolates between Gumbel and a particular case of Frechet distribution which arise in scaling limits of large and small $a$. My talk is based on joint work with Yan Fyodorov and Thomas Prellberg.