On completeness of random exponentials in the Bargmann-Fock space

We study the completeness/incompleteness properties of a system of exponent ials epsilon (Lambda)={e(pi lambdaz); lambda is an element of Lambda}, view ed as elements of the Bargmann-Fock space of entire functions. We assume th at the index set Lambda is a realization of a random point field in C (the support of a random measure). We prove that the properties are determined b y the density of the field, i.e., by the mean number of the field points pe r unit area. We also discuss certain implications and motivations of our re sults, in particular, the jumps of the integrated density of states of the Landau Hamiltonian with the random potential, equal to the sum of point sca tters. (C) 2001 American Institute of Physics.