Determinantal processes and completeness of random exponentials: the critical case

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For a locally finite point set Λ⊂ R, consider the collection of exponential functions given by EΛ:= {eⁱ λˣ: λ∈ L }. We examine the question whether EΛ spans the Hilbert space L²[-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 EΛ is indeed complete. We also answer an analogous question on 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 Zᵈ.