Given a nondecreasing sequence Λ={λn>0} such that limn→∞λn=∞, we consider the sequence NΛ:={λneiθn,n∈N}, where θn are independent random variables uniformly distributed on [0,2π]. We discuss the conditions on the sequence Λ under which NΛ is a zero set (a uniqness set) of a given weighted Fock space almost surely. The critical density of the sequence Λ with respect to the weight is found.