Abstract: 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.
This work is joint with Manjunath Krishnapur and Par Kurlberg.

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