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Determinantal processes and completeness of random exponentials: the critical case. (English) Zbl 1334.60083

Summary: 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 \(\mathbb {Z}^d\).

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
42C30 Completeness of sets of functions in nontrigonometric harmonic analysis
60G57 Random measures
60G60 Random fields
42C15 General harmonic expansions, frames
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
82B05 Classical equilibrium statistical mechanics (general)
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[1] Anderson, G., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2009) · Zbl 1184.15023 · doi:10.1017/CBO9780511801334
[2] Chistyakov, G., Lyubarskii, Y.: Random perturbations of exponential Riesz bases in \[L^2(-\pi,\pi )\] L2(-π,π). Ann. de l’institute Fourier. tome 47(1), 201-255 (1997) · Zbl 0860.42023 · doi:10.5802/aif.1565
[3] Chistyakov, G., Lyubarskii, Yu., Pastur, L.: On the completeness of random exponentials in the Bargmann-Fock space. J. Math. Phys. 42(8), 3754-3768 (2001) · Zbl 1009.42005
[4] Daley, D.J., Vere Jones, D.: An Introduction to the Theory of Point Processes, vol. I, II. Springer, New York (1997) · Zbl 1026.60061
[5] Goldman, A.: The Palm measure and the Voronoi tessellation for the Ginibre process. Ann Appl Probab 20(1), 90-128 (2010) · Zbl 1197.60047 · doi:10.1214/09-AAP620
[6] Ghosh, S., Peres, Y.: Rigidity and tolerance in point processes: Gaussian zeroes and Ginibre eigenvalues. arXiv:1211.2381v2 · Zbl 1405.60067
[7] Hough, J.B., Krishnapur, M., Peres, Y., Virag, B.: Zeros of Gaussian Analytic Functions and Determinantal Point Processes. AMS, Providence, Rhode Island (2010) · Zbl 1190.60038
[8] Holroyd, A.E., Soo, T.: Insertion and deletion tolerance of point processes (2010). arXiv:1007.3538v2 · Zbl 1291.60101
[9] Kallenberg, O.: Random Measures. Academic Press Inc., New York (1983) · Zbl 0544.60053
[10] Lyons, R.: Determinantal probability measures. Publ. Math. Inst. Hautes Etudes Sci. 98, 167-212 (2003) · Zbl 1055.60003 · doi:10.1007/s10240-003-0016-0
[11] Lyons, R.: Determinantal probability: basic properties and conjectures. In: Proceedings of International Congress of Mathematicians, Seoul, Korea, to appear (2014) · Zbl 0358.42007
[12] Lyons, R., Steif, J.: Stationary determinantal processes: phase multiplicity, Bernoullicity, entropy, and domination. Duke Math. J. 120(3), 515-575 (2003) · Zbl 1068.82010 · doi:10.1215/S0012-7094-03-12032-3
[13] Morris, B.: The components of the wired spanning forest are recurrent. Prob. Theor. Rel. Fields 125, 259-265 (2003) · Zbl 1031.60035 · doi:10.1007/s00440-002-0236-0
[14] Redheffer, R.: Completeness of sets of complex exponentials. Adv. Math. 24, 1-62 (1977) · Zbl 0358.42007 · doi:10.1016/S0001-8708(77)80002-9
[15] Redheffer, R., Young, R.: Completeness and basis properties of complex exponentials. Trans. Am. Math. Soc. 277(1), 93-111 (1983) · Zbl 0513.42011
[16] Seip, K.: Density theorems for sampling and interpolation in the Bargmann-Fock space. Bull. Am. Math. Soc. 26(2), 322-328 (1992) · Zbl 0808.30019 · doi:10.1090/S0273-0979-1992-00290-2
[17] Soshnikov, A.: Determinantal random point fields. Russian Math. Surveys (Uspekhi Mat. Nauk 55:5 107160) 55(5), 923-975 (2000) · Zbl 0991.60038
[18] Seip, K., Ulanovskii, A.: The Beurling-Malliavin density of a random sequence. Proc. Am. Math. Soc. 125(6), 1745-1749 (1997) · Zbl 0914.42005 · doi:10.1090/S0002-9939-97-03750-7
[19] Shirai, T., Takahashi, Y.: Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point processes. J. Funct. Anal. 205, 414-463 (2003) · Zbl 1051.60052 · doi:10.1016/S0022-1236(03)00171-X
[20] Yogeshwaran, D.: Negative association of point processes, preliminary version. https://sites.google.com/site/yogeshacademics/home/publications
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