Non-Universality of Nodal Length Distribution for Arithmetic Random Waves
Abstract
"Arithmetic random waves" are the Gaussian Laplace eigenfunctions on the two-dimensional torus (Rudnick and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013)). 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-It\^o chaos expansion for the nodal length shows that it is dominated by its th 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.
Keywords
Cite
@article{arxiv.1508.00353,
title = {Non-Universality of Nodal Length Distribution for Arithmetic Random Waves},
author = {Domenico Marinucci and Giovanni Peccati and Maurizia Rossi and Igor Wigman},
journal= {arXiv preprint arXiv:1508.00353},
year = {2017}
}
Comments
Accepted for publication on Geometric and Functional Analysis