Pär Kurlberg:Nodal length statistics for arithmetic random waves and probability measures arising from lattice points on circles.

Abstract. The Laplacian acting on the standard two dimensional torus has
spectral multiplicities related to the number of ways an integer can
be written as a sum of two integer squares.  Using these
multiplicities we can endow each eigenspace with a Gaussian
probability measure. This induces a notion of a random eigenfunction
(aka "random wave") on the torus, and we study the statistics of the
lengths of nodal sets (i.e., the zero set) of the eigenfunctions in
the "high energy limit".  In particular, we determine the variance for
a generic sequence of energy levels, and also find that the variance
can be different for certain "degenerate" subsequences; these
degenerate subsequences are closely related to circles on which
lattice points are very badly distributed.  In fact, the variance can
be described as a Fourier coefficient of certain probability measures
on the unit circle, namely measures that "come from" the intersection
of the Z^2 lattice with circles.