  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
non-universal, and is intimately related to the arithmetic of lattice points
lying on a circle with radius corresponding to the energy.
