Zeros distribution of gaussian entire functions

Abstract

In this paper we consider a random entire function of the form $f(z,\omega )={\sum }_{n=0}^{+\infty }{\xi }_n(\omega )a_n{z}^n,$ where ${\xi }_n(\omega )$ are independent standard\break complex gaussian random variables and $a_n\in \mathbb{C}$ satisfy the relations\break $\underset{n\to +\infty }{\overline{\lim }}\sqrt[n]{|a_n|}=0$ and $\#\{n:a_n\ne 0\}=+\infty .$ We investigate asymptotic properties of the probability $P_0(r)=P\{\omega :f(z,\omega )$ has no zeros inside $r\mathbb{D}\}.$ Denote $p_0(r)={\ln }^{-}{P}_0(r),N(r)=\#\{n:\ln (|a_n|r^n)>0\},$ $s(r)={\sum }_{n=0}^{+\infty }{\ln }^{+}(|a_n|r^n).$ Assuming that $a_0\ne 0$ we prove that $0\le {\underline{\lim }}_{r\to +\infty ,r\notin E}\frac{\ln (p_0(r)-s(r))}{\ln s(r)},{\overline{\lim }}_{r\to +\infty ,r\notin E}\frac{\ln (p_0(r)-s(r))}{\ln s(r)}\le \frac{1}{2},$ $\underset{r\to +\infty ,r\notin E}{\lim }\frac{\ln (p_0(r)-s(r))}{\ln N(r)}=1.$ where $E$ is a set of finite logarithmic measure. Remark that the previous inequalities are sharp. Also we give an answer to open question from \cite[p. 119]{nishry 5}.

References in corpus (3)

• Topics in the Value Distribution of Random Analytic Functions
• Random complex zeroes, I. Asymptotic normality
• Random complex zeroes, III. Decay of the hole probability