Restriction of 3D arithmetic Laplace eigenfunctions to a plane

Maffucci, Riccardo W.

doi:10.1214/20-ejp457

Electron. J. Probab.

2020

DOI: 10.1214/20-ejp457

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Restriction of 3D arithmetic Laplace eigenfunctions to a plane

Riccardo W. Maffucci¹

Abstract: We consider a random Gaussian ensemble of Laplace eigenfunctions on the 3D torus, and investigate the 1-dimensional Hausdorff measure ('length') of nodal intersections against a smooth 2-dimensional toral sub-manifold ('surface'). A prior result of ours prescribed the expected length, universally proportional to the area of the reference surface, times the wavenumber, independent of the geometry.In this paper, for surfaces contained in a plane, we give an upper bound for the nodal intersection length variance,… Show more

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“…Results on the intersection of nodal sets against a surface can be found in ; , see also Maffucci (2020) for a study of the intersection length obtained when intersecting nodal sets of ARWs with planes.…”

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confidence: 99%

Fluctuations of nodal sets on the 3-torus and general cancellation phenomena

Notarnicola¹

2021

ALEA

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Benatar and Maffucci (2019) established an asymptotic law for the variance of the nodal surface of arithmetic random waves on the 3-torus in the high-energy limit. In a subsequent work, Cammarota (2019) proved a universal non-Gaussian limit theorem for the nodal surface. In this paper, we study the nodal intersection length and the number of nodal intersection points associated, respectively, with two and three independent arithmetic random waves of same frequency on the 3-torus. For these quantities, we compute their expected value, asymptotic variance as well as their limiting distribution. Our results are based on Wiener-Itô expansions for the volume and naturally complement the findings of Cammarota (2019). At the core of our analysis lies an abstract cancellation phenomenon applicable to the study of level sets of arbitrary Gaussian random fields, that we believe has independent interest.

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Fluctuations of nodal sets on the 3-torus and general cancellation phenomena

Notarnicola¹

2021

ALEA

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Asymptotic distribution of nodal intersections for ARW against a surface

Maffucci,

Rossi

2024

Journal of Mathematical Physics

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We investigate Gaussian Laplacian eigenfunctions (Arithmetic Random Waves) on the three-dimensional standard flat torus, in particular the asymptotic distribution of the nodal intersection length against a fixed regular reference surface. Expectation and variance have been addressed by Maffucci [Ann. Henri Poincaré 20(11), 3651–3691 (2019)] who found that the expected length is proportional to the square root of the eigenvalue times the area of the surface, while the asymptotic variance only depends on the geometry of the surface, the projected lattice points being equidistributed on the two-dimensional unit sphere in the high-energy limit. He also noticed that there are “special” surfaces, so-called static, for which the variance is of smaller order; however he did not prescribe the precise asymptotic law in this case. In this paper, we study second order fluctuations of the nodal intersection length. Our first main result is a Central Limit Theorem for “generic” surfaces, while for static ones, a sphere or a hemisphere e.g., our main results are a non-Central Limit Theorem and a precise asymptotic law for the variance of the nodal intersection length, conditioned on the existence of so-called well-separated sequences of Laplacian eigenvalues. It turns out that, in this regime, the nodal area investigated by Cammarota [Trans. Am. Math. Soc. 372(5), 3539–3564 (2019)] is asymptotically fully correlated with the length of the nodal intersections against any sphere. The main ingredients for our proofs are the Kac-Rice formula for moments, the chaotic decomposition for square integrable functionals of Gaussian fields, and some arithmetic estimates that may be of independent interest.

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Restriction of 3D arithmetic Laplace eigenfunctions to a plane

Cited by 2 publications

References 39 publications

Fluctuations of nodal sets on the 3-torus and general cancellation phenomena

Fluctuations of nodal sets on the 3-torus and general cancellation phenomena

Asymptotic distribution of nodal intersections for ARW against a surface

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