Determinantal processes and completeness of random exponentials: the critical case

By Subhro Ghosh. 2012

94 citations recorded by the source corpus.

For a locally finite point set $$\Lambda \subset \mathbb {R}$$Λ⊂R, consider the collection of exponential functions given by $$\mathcal {E}_{\Lambda }:=\{ e^{i \lambda x } : \lambda \in \Lambda \}$$EΛ:={eiλx:λ∈Λ}. We examine the question whether $$\mathcal {E}_{\Lambda }$$EΛ spans the Hilbert space $$L^2[-\pi ,\pi ]$$L2[-π,π], when $$\Lambda $$Λ is random. For several point processes of interest, this belongs to a certain critical case of the corresponding question for deterministic $$\Lambda $$Λ, about which little is known. For $$\Lambda $$Λ the continuum sine kernel process, obtained as the bulk limit of GUE eigenvalues, we establish that $$\mathcal {E}_{\Lambda }$$EΛ is indeed complete almost surely. We also answer an analogous question on $$\mathbb {C}$$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 $$\mathbb {Z}^d$$Zd.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/37f59da81b5eaabbcacfa7fb3877fcee6e199d4f

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In Mathematical Physics

In Statistics and Probability

Citations in the corpus

On the topology of random complexes built over stationary point processes. (2012, background): “Stationary determinantal point processes with continuous kernels are negatively associated [18], Corollary 6.”