Fluctuations in random complex zeroes: Asymptotic normality revisited
Open accessAbstract
These results complement our recent results from “Clustering of correlation functions for random complex zeroes”, where, using different methods, we prove that the fluctuations of n(R,h) are asymptotically normal when h is bounded, and the variance of n(R,h) grows at least as a positive power of R.
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Research works printed but not linked: 21
- S. Bernstein, Sur l'extension du théoréme limite du calcul des probabilités aux sommes de quantités dépendantes. Math. Ann. 97 (1927), 1-59.
- P. Bleher, B. Shiffman, S. Zelditch Universality and scaling of correlations between zeros on complex manifolds, Invent. Math. 142 (2000), 351-395. arXiv: math-ph/9904020 · arXiv
- E. Bogomolny, O. Bohigas, P. Leboeuf, Distribution of roots of random polynomials. Phys. Rev. Lett. 68 (1992), 2726-2729;
- Quantum chaotic dynamics and random polynomials. J. Statist. Phys. 85 (1996), 639-679. arXiv: chao-dyn/9604001 · arXiv
- P. J. Forrester and G. Honner, Exact statistical properties of the zeros of complex random polynomials. J. Phys. A 32 (1999), 2961-2981. arXiv: cond-mat/9811142 · arXiv
- S. Janson, Gaussian Hilbert Spaces, Cambridge University Press, 1997.
- J. H. Hannay, Chaotic analytic zero points: exact statistics for those of a random spin state. J. Phys. A 29 (1996), L101-L105; The chaotic analytic function. J. Phys. A 31 (1998), L755-L761.
- G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, 1934.
- E. Kostlan, Random polynomials and the statistical fundamental theorem of algebra, Preprint, 1987. Available at www.developmentserver.com/ randompolynomials/
- M. Krishnapur, Zeros of Random Analytic Functions, Ph.D. Thesis, Berkeley, 2006. arXiv:math/0607504v1 · arXiv
- F. Nazarov, M. Sodin, A. Volberg, Transportation to random zeroes by the gradient flow, Geom. and Funct. Anal. 17 (2007), 887-935. arXiv: math.CV/0510654 · arXiv
- F. Nazarov, M. Sodin, A. Volberg, The Jancovici -Lebowitz -Manificat law for large fluctuations of random complex zeroes, Comm. Math. Phys. 284 (2008), 833-865. arXiv:0707.3863 · arXiv
- B. Rider and B. Virág, The noise in the circular law and the Gaussian free field, Int. Math. Res. Not. IMRN, no 2 (2007), arXiv:math/0606663; · arXiv
- Complex determinantal processes and H 1 noise, Electron. J. Probab. 12 (2007), 1238-1257, arXiv:math/0608785 · arXiv
- B. Shiffman and S. Zelditch, Number variance of random zeros on complex manifold. Geom. and Funct. Anal., to appear. arXiv: math.CV/0608743 · arXiv
- M. Sodin, Zeros of Gaussian Analytic Functions, Math. Res. Lett. 7 (2000), 371-381. arXiv:math/0007030 · arXiv
- M. Sodin and B. Tsirelson, Random complex zeroes. I. Asympotic normality, Israel J. Math. 144 (2004), 125-149;
- II. Perturbed Lattice, ibid 152 (2006), 105-124;
- III. Decay of the hole probability, ibid 147 (2005), 371-379. arxiv: math.CV/0210090, math.CV/0309449, and math.CV/0312258 · arXiv
- A. Soshnikov, Gaussian Limit for Determinantal Random Point Fields. Annals of Probab. 30 (2002), 171-187. arXiv:math/0006037 · arXiv
- B. Tsirelson, Moderate deviations for random fields and random complex zeroes, arXiv:0801.1050v1. · arXiv
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Lachièze-Rey, R. (2021) · arXiv · doi:10.48550/arxiv.2104.07290 · Cited by 1 · see citing works
“In higher dimensions, zeros of Gaussian entire functions [14, 25] and nodal sets of high enery Gaussian harmonics on a compact manifold [19, 36, 22] (and their Euclidean counterpart the Random Wave Model [24]) have attracted a lot of attention from both physicists and mathematicians.”
Background
- Authors
- Nazarov, F., & Sodin, M.
- Affiliations
- University of Wisconsin–Madison (US)
- Tel Aviv University (IL)
- Corresponding author
- Sodin, M.
- ORCID
- Venue
- International Mathematics Research Notices
- Publisher
- Oxford University Press (OUP)
- Published
- 2011
- Volume
- 2011
- Issue
- 24
- Pages
- 5720-5759
- ISSN
- 1073-7928 (ISSN-L)
- 1687-0247
- OpenAlex
- W1792499074
- Cited by
- 55
- References
- 0
- Citations by year
- 2026: 3 · 2025: 1 · 2024: 1 · 2023: 4 · 2022: 3 · 2021: 6 · 2020: 5 · 2019: 6 · 2018: 4 · 2017: 4 · 2016: 4 · 2015: 3 · 2014: 5 · 2013: 2 · 2012: 1
- FWCI
- 3.08
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- International Mathematics Research Notices (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Random Matrices and Applications
- Physical Sciences › Mathematics › Algebra and Number Theory › Analytic Number Theory Research
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical functions and polynomials
- Keywords
- asymptotic normality, fluctuations, counting functions, correlation functions, clustering, random analytic functions
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