Journal title
International Mathematics Research Notices
Last updated
2019-08-26T21:42:17.707+01:00
Abstract
We consider the ensemble of random Gaussian Laplace eigenfunctions on
$\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$ (`$3d$ 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.
$\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$ (`$3d$ 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.
Symplectic ID
733156
Download URL
http://arxiv.org/abs/1708.07015v1
Submitted to ORA
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Publication type
Journal Article