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Non-Universality of Nodal Length Distribution for Arithmetic Random Waves
Open accessAbstract
“Arithmetic random waves” are the Gaussian Laplace eigenfunctions on the two-dimensional torus (Rudnick and Wigman in Annales de l’Insitute Henri Poincaré 9(1):109–130, 2008; Krishnapur et al. in Annals of Mathematics (2) 177(2):699–737, 2013). In this paper we find that their nodal length converges to a non-universal (non-Gaussian) limiting distribution, depending on the angular distribution of lattice points lying on circles. Our argument has two main ingredients. An explicit derivation of the Wiener–Itô chaos expansion for the nodal length shows that it is dominated by its 4th order chaos component (in particular, somewhat surprisingly, the second order chaos component vanishes). The rest of the argument relies on the precise analysis of the fourth order chaotic component.
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Cites
Nodal length fluctuations for arithmetic random waves
Krishnapur, M., Kurlberg, P., & Wigman, I. (2013) · Annals of Mathematics · doi:10.4007/annals.2013.177.2.8 · Cited by 89
Self-citation
Research works printed but not linked: 18
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- Marinucci D., Rossi M.: Stein–Malliavin approximations for nonlinear functionals of random eigenfunctions on S d . Journal of Functional Analysis, 268(8), 2379–2420 (2015) · DOI
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Cited by
Diophantine Gaussian excursions and random walks
Lachièze-Rey, R. (2021) · arXiv · doi:10.48550/arxiv.2104.07290 · Cited by 1
“Such random Gaussian harmonics have been recently heavily studied in dimension 2 in the mathematics literature, especially on the sphere or the torus [19, 23, 12, 22], and the Euclidean version Xd has also been investigated, through its percolation properties [24] or the statistical properties of their nodal lines [26…”
Background
Metadata
- Authors
- Marinucci, D., Peccati, G., Rossi, M., & Wigman, I.
- Affiliations
- University of Rome Tor Vergata (IT)
- University of Luxembourg (LU)
- King's College London (GB)
- Corresponding author
- Marinucci, Domenico
- Venue
- Geometric and Functional Analysis
- Published
- 2016-06
- Volume
- 26
- Issue
- 3
- Pages
- 926-960
- ISSN
- 1016-443X (ISSN-L)
- 1420-8970
- OpenAlex
- W2271093040
- Cited by
- 12
- References
- 1
- Citations by year
- 2025: 1 · 2024: 3 · 2023: 1 · 2022: 2 · 2020: 1 · 2019: 1 · 2018: 1 · 2016: 2
- FWCI
- 2.32
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Geometric and Functional Analysis (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Open Repository and Bibliography (University of Luxembourg) (submittedVersion, other-oa)
- CINECA IRIS Institutial research information system (University of Pisa) (submittedVersion)
- Cineca Institutional Research Information System (Tor Vergata University) (submittedVersion)
- BOA (University of Milano-Bicocca) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Mathematics › Algebra and Number Theory › Analytic Number Theory Research
- Physical Sciences › Mathematics › Mathematical Physics › Spectral Theory in Mathematical Physics
- Physical Sciences › Mathematics › Numerical Analysis › Mathematical Approximation and Integration
- Keywords
- two-dimensional torus, non-Gaussian limit, irregular waves
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