Random analytic functions with a prescribed growth rate in the unit disk

Abstract

Let $\mathcal {R}f$ be the randomization of an analytic function over the unit disk in the complex plane $$ \begin{align*}\mathcal{R} f(z) =\sum_{n=0}^\infty a_n X_n z^n \in H({\mathbb D}), \end{align*} $$where $f(z)=\sum _{n=0}^\infty a_n z^n \in H({\mathbb D})$ and $(X_n)_{n \geq 0}$ is a standard sequence of independent Bernoulli, Steinhaus, or complex Gaussian random variables. In this paper, we demonstrate that prescribing a polynomial growth rate for random analytic functions over the unit disk leads to rather satisfactory characterizations of those $f \in H({\mathbb D})$ such that ${\mathcal R} f$ admits a given rate almost surely. In particular, we show that the growth rate of the random functions, the growth rate of their Taylor coefficients, and the asymptotic distribution of their zero sets can mutually, completely determine each other. Although the problem is purely complex analytic, the key strategy in the proofs is to introduce a class of auxiliary Banach spaces, which facilitate quantitative estimates.