  Given a nondecreasing sequence $\Lambda=\{\lambda_n>0\}$ such that
$\displaystyle\lim_{n\to\infty} \lambda_n=\infty,$ we consider the sequence
$\mathcal N_\Lambda:=\left\{\lambda_ne^{i\theta_n},n\in\,\mathbb N\right\}$,
where $\theta_n$ are independent random variables uniformly distributed on
$[0,2\pi].$ We discuss the conditions on the sequence $\Lambda$ under which
$\mathcal N_\Lambda$ is a zero set (a uniqness set) of a given weighted Fock
space almost surely. The critical density of the sequence $\Lambda$ with
respect to the weight is found.
