Determinantal processes and completeness of random exponentials: the critical case

For a locally finite point set $\mathrm{\Lambda }\subset \mathbb{R}$, consider the collection of exponential functions given by $$\mathcal{E}${\Lambda}:= {e^{i \lambda x} : \lambda \in L }$\mathrm{.}Weexaminethequestionwhether$$\mathcal{E}${\Lambda}$spanstheHilbertspace$L^2[-\pi,\pi]$,when$\Lambda$ is random. For several point processes of interest, this belongs to a certain critical case of the corresponding question for deterministic $\mathrm{\Lambda }$, about which little is known. For $\mathrm{\Lambda }$ the continuum sine kernel process, obtained as the bulk limit of GUE eigenvalues, we establish that ${\mathcal{E}}_{\mathrm{\Lambda }}$ 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 ${\mathbb{Z}}^d$.