Determinantal processes and completeness of random exponentials: the critical case

For a locally finite point set $Λ\subset \mathbb{R}$, consider the collection of exponential functions given by $\mathcal{E}_Λ:= \{e^{i λx} : λ\in L \}$. We examine the question whether $\mathcal{E}_Λ$ spans the Hilbert space $L^2[-π,π]$, when $Λ$ is random. For several point processes of interest, this belongs to a certain critical case of the corresponding question for deterministic $Λ$, about which little is known. For $Λ$ the continuum sine kernel process, obtained as the bulk limit of GUE eigenvalues, we establish that $\mathcal{E}_Λ$ 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 $\mathbb{Z}^d$.