The probability of absence zeros in the disc for some random analytic functions

Анотація

We consider random analytic function $f(z;\omega) =\sum_{n=0}^{+\infty}\phi_na_nz^n$ where $a_n > 0$ such that $\lim_{n\to+\infty}\sqrt[n]{a_n} = 1$, $\phi_n=\phi_n(\omega)$ are standard complex gaussian random variables in probability space of Steinhaus $(\Omega,\mathcal A,P)$. If $\{a_n\}$ is log-concave and $\lim_{r\to 1-0}\ln\ln M(r)=+\infty$, then $(\exists E(\varepsilon,f)=E\subset(0; 1))(\forall r\in (0; 1)\setminus E)$ $\overline{\lim}_{r\to 1-0}\frac1{1-r}\mathrm{meas}(E\cap [r,1))=0$ and $\ln^- P(\{\omega:f(z;\omega)\neq 0\})=S(r)+o(S(r))$ $(r\to 1-0)$,  where $S(r) = 2 \sum_{n:a_nr^n\ge 1}\ln(a_nr^n)$.