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Random Waves On <span class="MathJax_Preview">\mathbb T^3</span><script type="math/tex" id="MathJax-Element-1">\mathbb T^3</script>: Nodal Area Variance and Lattice Point Correlations | OUP Journals & Magazine | IEEE Xplore

Random Waves On \mathbb T^3: Nodal Area Variance and Lattice Point Correlations

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Abstract:

We consider the ensemble of random Gaussian Laplace eigenfunctions on \mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3 (“3 dimensional arithmetic random waves”), and study the di...Show More

Abstract:

We consider the ensemble of random Gaussian Laplace eigenfunctions on \mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3 (“3 dimensional arithmetic random waves”), and study the distribution of their nodal surface area. The expected area is proportional to the square root of the eigenvalue, or “energy”, of the eigenfunction. We show that the nodal area variance obeys an asymptotic law. The resulting asymptotic formula is closely related to the angular distribution and correlations of lattice points lying on spheres.
Published in: International Mathematics Research Notices ( Volume: 2019, Issue: 10, May 2019)
Page(s): 3032 - 3075
Date of Publication: May 2019

ISSN Information: