A sufficient condition for random zero sets of Fock spaces

Abstract

Let $(r_n)_{n=1}^\infty$ be a non-decreasing sequence of radii in $(0, \infty )$, and let $(\theta _n)_{n=1}^\infty$ be a sequence of independent random arguments uniformly distributed in $[0, 2\pi )$. In this paper, we establish a new sufficient condition on the sequence $(r_n)_{n=1}^\infty$ under which $(r_ne^{i\theta _n})_{n=1}^\infty$ is almost surely a zero set for Fock spaces. The condition is in terms of the sum of two characteristics involving the counting function. The sharpness of this condition is discussed and examples are presented to illustrate it.