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 eigenfunctions
on the torus ("arithmetic random waves''.)  We study the
distribution of the nodal length of random Laplace eigenfunctions for high
eigenvalues,and our primary result is that the asymptotics for the variance is
non-universal, and is intimately related to the arithmetic of
lattice points lying on a circle with radius corresponding to the
energy. This work is joint with Manjunath Krishnapur and Par Kurlberg