  We consider Berry's random planar wave model (1977) for a positive Laplace
eigenvalue $E>0$, both in the real and complex case, and prove limit theorems
for the nodal statistics associated with a smooth compact domain, in the
high-energy limit ($E\to \infty$). Our main result is that both the nodal
length (real case) and the number of nodal intersections (complex case) verify
a Central Limit Theorem, which is in sharp contrast with the non-Gaussian
behaviour observed for real and complex arithmetic random waves on the flat
$2$-torus, see Marinucci et al. (2016) and Dalmao et al. (2016). Our findings
can be naturally reformulated in terms of the nodal statistics of a single
random wave restricted to a compact domain diverging to the whole plane. As
such, they can be fruitfully combined with the recent results by Canzani and
Hanin (2016), in order to show that, at any point of isotropic scaling and for
energy levels diverging sufficently fast, the nodal length of any Gaussian
pullback monochromatic wave verifies a central limit theorem with the same
scaling as Berry's model. As a remarkable byproduct of our analysis, we
rigorously confirm the asymptotic behaviour for the variances of the nodal
length and of the number of nodal intersections of isotropic random waves, as
derived in Berry (2002).
