Determinantal processes and completeness of random exponentials: the critical case

Abstract

For a locally finite point set Λ ⊂ R, consider the collection of exponential functions given by E Λ := {e iλx : λ ∈ Λ}. We examine the question whether E Λ spans the Hilbert space L 2 [−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 E Λ is indeed complete almost surely. We also answer an analogous question on 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 Z d .

Rigidity of determinantal point processes with the Airy, the Bessel and the Gamma kernel

“…For the sine-process, rigidity is due to Ghosh [6]. For the Ginibre ensemble, rigidity has been established by Ghosh and Peres [7]; see also Osada and Shirai [16].…”

“…We recall the sufficient condition for rigidity of a point process given by Ghosh [6], Ghosh and Peres [7]. [6], Ghosh and Peres [7]) Let P be a Borel probability measure on Conf(M).…”

“…Given a Borel subset C ⊂ M, we let F C be the σ -algebra generated by all random variables of the form # B , B ⊂ C. If P is a point process on M, then we write F P C for the P-completion of F C . The following definition of rigidity of a point process is due to Ghosh [6] (cf. also Ghosh and Peres [7]).…”

“…[6], Ghosh and Peres [7]) Let P be a Borel probability measure on Conf(M). Assume that for any ε > 0 and any bounded subset B ⊂ M there exists a bounded measurable function f of bounded support such that f ≡ 1 on B and Var P S f < ε.…”

Discrete Approximations of Determinantal Point Processes on Continuous Spaces: Tree Representations and Tail Triviality

“…Determinantal point processes on discrete spaces have a well-behaved algebraic structure; as a result, some important facts are only known for discrete determinantal point processes [4,[6][7][8]12]. One such example is tail triviality, which says that each event of a tail σ -field Tail(S) takes value 0 or 1.…”