Determinantal processes and completeness of random exponentials: the critical case

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 $\mathbb{Z}^d$.