Determinantal processes and completeness of random exponentials: the critical case

Abstract

For a locally finite point set \(\Lambda \subset \mathbb {R}\), consider the collection of exponential functions given by \(\mathcal {E}_{\Lambda }:=\{ e^{i \lambda x } : \lambda \in \Lambda \}\). 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 almost surely. We also answer an analogous question on \(\mathbb {C}\) for the Ginibre ensemble, arising as weak limits of the spectra of certain non-Hermitian Gaussian random matrices. In fact we establish completeness for any “rigid” determinantal point process in a general setting. In addition, we partially answer two questions of Lyons and Steif about stationary determinantal processes on \(\mathbb {Z}^d\).