  For a locally finite point set $\Lambda \subset \mathbb{R}$, consider the
collection of exponential functions given by $\mathcal{E}_{\Lambda}:= \{e^{i
\lambda x} : \lambda \in L \}$. We examine the question whether
$\mathcal{E}_{\Lambda}$ spans the Hilbert space $L^2[-\pi,\pi]$, when $\Lambda$
is random. For several point processes of interest, this belongs to a certain
critical case of the corresponding question for deterministic $\Lambda$, about
which little is known. For $\Lambda$ the continuum sine kernel process,
obtained as the bulk limit of GUE eigenvalues, we establish that
$\mathcal{E}_{\Lambda}$ is indeed complete. We also answer an analogous
question on $\mathbb{C}$ for the Ginibre ensemble, arising as weak limits of
certain non-Hermitian random matrix eigenvalues. In fact we establish
completeness for any "rigid" determinantal point process in a general setting.
In addition, we partially answer two questions due to Lyons and Steif about
stationary determinantal processes on $\mathbb{Z}^d$.
