On the Volume of Nodal Sets for Eigenfunctions of the Laplacian on the Torus

"Arithmetic random waves" are the Gaussian Laplace eigenfunctions on the twodimensional torus [RW, KKW]. 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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“…(a rigorous manifistation of (3.26)), and moreover, by [RW,Lemma 3.2], L ε n is uniformly bounded, that is:…”

Section: Approximating the Nodal Lengthmentioning

confidence: 99%

Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

et al. 2016

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“…A beautiful topic that we do not discuss here is the zero sets of random eigenfunctions (linear combinations with random coefficients). The interested reader can start wtih [77], [81], [18], [93], [51] for the introduction to random eigenfunctions.…”

Section: Two Examplesmentioning

confidence: 99%

Review of Yau’s conjecture on zero sets of Laplace eigenfunctions

Logunov

Malinnikova

2018

Current Developments in Mathematics

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This is a review of old and new results and methods related to the Yau conjecture on the zero set of Laplace eigenfunctions. The review accompanies two lectures given at the conference CDM 2018. We discuss the works of Donnelly and Fefferman including their solution of the conjecture in the case of real-analytic Riemannian manifolds. The review exposes the new results for Yau's conjecture in the smooth setting. We try to avoid technical details and emphasize the main ideas of the proof of Nadirashvili's conjecture. We also discuss two-dimensional methods to study zero sets.

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“…Lemma 2.12 (Lemma 3.3 from [11]). Let g(t) be a trigonometric polynomial on [0, 2π] of degree at most M .…”

Section: 4mentioning

confidence: 99%

“…was studied by Rudnick and Wigman [11]. In this case, it is not difficult to see that the expectation is given by EZ(f T m ) = const · √ E. Moreover, they prove that as the eigenspace dimension N grows to infinity, the variance is bounded by…”

Section: Introductionmentioning

confidence: 98%

On the distribution of the nodal sets of random spherical harmonics

Wigman

2009

Journal of Mathematical Physics

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Abstract. We study the volume of the nodal set of eigenfunctions of the Laplacian on the m-dimensional sphere. It is well known that the eigenspaces corresponding to En = n(n + m − 1) are the spaces En of spherical harmonics of degree n, of dimension N . We use the multiplicity of the eigenvalues to endow En with the Gaussian probability measure and study the distribution of the m-dimensional volume of the nodal sets of a randomly chosen function. The expected volume is proportional to √ En. One of our main results is bounding the variance of the volume to be O(In addition to the volume of the nodal set, we study its Leray measure. We find that its expected value is n independent. We are able to determine that the asymptotic form of the variance is const N .

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On the Volume of Nodal Sets for Eigenfunctions of the Laplacian on the Torus

Cited by 87 publications

References 11 publications

Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

Review of Yau’s conjecture on zero sets of Laplace eigenfunctions

On the distribution of the nodal sets of random spherical harmonics

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