Concentration for nodal component count of Gaussian Laplace eigenfunctions

Priya, Lakshmi

doi:10.48550/arxiv.2012.10302

Cited by 2 publications

(3 citation statements)

References 20 publications

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“…Nevertheless, even in the oscillating case the attained bounds are best known. For instance, for the nodal set of the RPW the attained bound cR 7/2 improves the previously best-known bound cR 4−1/8 [38] (and moreover is valid at all levels). Remark 1.13.…”

Section: 22supporting

confidence: 51%

“…Note that this bound is only expected to be attained for very degenerate Gaussian fields (see [6,9] for examples). For the nodal level ℓ = 0 of the random plane wave, Priya [38] recently improved the upper bound to cR 4−1/8 . This bound is deduced from the exponential concentration of the nodal component count, and the proof does not extend to general levels/fields.…”

Section: Introductionmentioning

confidence: 99%

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Fluctuations of the number of excursion sets of planar Gaussian fields

2022

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For a smooth, stationary, planar Gaussian field, we consider the number of connected components of its excursion set (or level set) contained in a large square of area R 2 . The mean number of components is known to be of order R 2 for generic fields and all levels. We show that for certain fields with positive spectral density near the origin (including the Bargmann-Fock field), and for certain levels ℓ, these random variables have fluctuations of order at least R, and hence variance of order at least R 2 . In particular this holds for excursion sets when ℓ is in some neighbourhood of zero, and it holds for excursion/level sets when ℓ is sufficiently large. We prove stronger fluctuation lower bounds of order R α for α ∈ [1, 2] in the case that the spectral density has a singularity at the origin. Finally we show that the number of excursion/level sets for the random plane wave at certain levels has fluctuations of order at least R 3/2 , and hence variance of order at least R 3 . We expect that these bounds are of the correct order, at least for generic levels.

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Section: 22supporting

confidence: 51%

Section: Introductionmentioning

confidence: 99%

Fluctuations of the number of excursion sets of planar Gaussian fields

2022

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“…Note that this bound is only expected to be attained for very degenerate Gaussian fields (see [7,9] for examples). Various concentration bounds have also been established for N ⋆ (R, ℓ) [38,39,10], but these do not lead to improved bounds on the variance in the short-range correlated case. Related questions have also been studied in the 'sparse' regime ℓ R → ∞ as R → ∞ [42].…”

Section: Introductionmentioning

confidence: 99%

A central limit theorem for the number of excursion set components of Gaussian fields

Beliaev¹,

McAuley²,

Muirhead³

2022

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For a smooth stationary Gaussian field on R d and level ℓ ∈ R, we consider the number of connected components of the excursion set {f ≥ ℓ} (or level set {f = ℓ}) contained in large domains. The mean of this quantity is known to scale like the volume of the domain under general assumptions on the field. We prove that, assuming sufficient decay of correlations (e.g. the Bargmann-Fock field), a central limit theorem holds with volume-order scaling. Previously such a result had only been established for 'additive' geometric functionals of the excursion/level sets (e.g. the volume or Euler characteristic) using Hermite expansions. Our approach, based on a martingale analysis, is more robust and can be generalised to a wider class of topological functionals. A major ingredient in the proof is a third moment bound on critical points, which is of independent interest.

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Concentration for nodal component count of Gaussian Laplace eigenfunctions

Cited by 2 publications

References 20 publications

Fluctuations of the number of excursion sets of planar Gaussian fields

Fluctuations of the number of excursion sets of planar Gaussian fields

A central limit theorem for the number of excursion set components of Gaussian fields

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