Nodal length fluctuations for arithmetic random waves


Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus (“arithmetic random waves”). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is nonuniversal. Our result is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.


Manjunath Krishnapur

Indian Institute of Science, Bangalore, India

Pär Kurlberg

Royal Institute of Technology, Stockholm, Sweden

Igor Wigman

Cardiff University, Wales, UK

Current address:

King's College London, London, UK