Zeros of conditional Gaussian analytic functions, random sub-unitary matrices and q-series

Abstract

We investigate radial statistics of zeros of hyperbolic Gaussian Analytic Functions (GAF) of the form $\ddot{I}†(z)={\sum }_{k\ge 0}{c}_k{z}^k$ given that $|\ddot{I}†(0){|}^2=t$ and assuming coefficients $c_k$ to be independent standard complex normals. We obtain the full conditional distribution of $N_q$, the number of zeros of $\ddot{I}†(z)$ within a disk of radius $\sqrt{q}$ centred at the origin, and prove its asymptotic normality in the limit when $q\to 1^{-}$, the limit that captures the entire zero set of $\ddot{I}†(z)$. In the same limit we also develop precise estimates for conditional probabilities of moderate to large deviations from normality. Finally, we determine the asymptotic form of $P_k(t;q)=\mathrm{Prob}\{N_q=k||\ddot{I}†(0){|}^2=t\}$ in the limit when $k$ is kept fixed whilst $q$ approaches 1. To leading order, the hole probability $P_0(t;q)$ does not depend on $t$ for $t>0$ but yet is different from that of $P_0(t=0;q)$ and coincides with the hole probability for unconditioned hyperbolic GAF of the form ${\sum }_{k\ge 0}\sqrt{k+1}{c}_k{z}^k$. We also find that asymptotically as $q\to 1^{-}$, $P_k(t;q)=e^t{P}_k(0;q)$ for every fixed $k\ge 1$ with $P_k(0;q)=\mathrm{Prob}\{N_q=k-1\}$.

48 pages, 8 figures