  The Seba billiard, a rectangular torus with a point scatterer, is a popular
model to study the transition between integrability and chaos in quantum
systems. Whereas such billiards are classically essentially integrable, they
may display features such as quantum ergodicity [KU] which are usually
associated with quantum systems whose classical dynamics is chaotic. Seba
proposed that the eigenfunctions of toral point scatterers should also satisfy
Berry's random wave conjecture, which implies that the semiclassical moments of
the eigenfunctions ought to be Gaussian.
  We prove a conjecture of Keating, Marklof and Winn who suggested that Seba
billiards with irrational aspect ratio violate the random wave conjecture. More
precisely, in the case of diophantine tori, we construct a subsequence of
eigenfunctions of essentially full density and show that its semiclassical
moments cannot be Gaussian.
