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Zeros of conditional Gaussian analytic functions, random sub-unitary matrices and q-series (2412.06086v1)

Published 8 Dec 2024 in math.PR, math-ph, and math.MP

Abstract: We investigate radial statistics of zeros of hyperbolic Gaussian Analytic Functions (GAF) of the form $\varphi (z) = \sum_{k\ge 0} c_k zk$ given that $|\varphi (0)|2=t$ 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 $\varphi (z)$ 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 $\varphi (z)$. 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 $P_k(t;q)=\mathrm{Prob} { N_q= k | |\varphi(0)|2=t }$ 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 $\sum_{k\ge 0} \sqrt{k+1}\, c_k zk$. We also find that asymptotically as $q \to 1{-}$, $P_k(t;q)= et P_{k}(0;q)$ for every fixed $k \ge 1$ with $P_{k}(0;q)= \mathrm{Prob} { N_q =k-1 }$.

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