Zeros of conditional Gaussian analytic functions, random sub-unitary matrices and q-series
We investigate radial statistics of zeros of hyperbolic Gaussian Analytic Functions (GAF) of the form $\varphi (z) = \sum_{k\ge 0} c_k z^k$ given that $|\varphi (0)|^2=t$ and assuming coefficients $c_k$ to be independent standard complex normals. We obtain the full conditional distribution of $N_q$,...
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Zusammenfassung: | We investigate radial statistics of zeros of hyperbolic Gaussian Analytic
Functions (GAF) of the form $\varphi (z) = \sum_{k\ge 0} c_k z^k$ 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 $\sum_{k\ge 0} \sqrt{k+1}\, c_k z^k$. 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 $P_{k}(0;q)= \mathrm{Prob} \{ N_q =k-1 \}$. |
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DOI: | 10.48550/arxiv.2412.06086 |