We consider the ensemble of random Gaussian Laplace eigenfunctions on T3 = R3/Z3 ("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.
Random waves on T3: Nodal area variance and lattice point correlations
Maffucci R. W.
2019-01-01
Abstract
We consider the ensemble of random Gaussian Laplace eigenfunctions on T3 = R3/Z3 ("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.
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