  "Arithmetic random waves" are the Gaussian Laplace eigenfunctions on the
two-dimensional torus (Rudnick and Wigman (2008), Krishnapur, Kurlberg and
Wigman (2013)). In this paper we find that their nodal length converges to a
non-universal (non-Gaussian) limiting distribution, depending on the angular
distribution of lattice points lying on circles. Our argument has two main
ingredients. An explicit derivation of the Wiener-It\^o chaos expansion for the
nodal length shows that it is dominated by its $4$th order chaos component (in
particular, somewhat surprisingly, the second order chaos component vanishes).
The rest of the argument relies on the precise analysis of the fourth order
chaotic component.
