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 $φ(z) = \sum_{k\ge 0} c_k z^k$ given that $|φ(0)|^2=t$ 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 $φ(z)$ 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 $φ(z)$. 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 $P_k(t;q)=\mathrm{Prob} \{ N_q= k | |φ(0)|^2=t \}$ 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 $P_{k}(0;q)= \mathrm{Prob} \{ N_q =k-1 \}$.