We consider Berry's random planar wave model (1977) for a positive Laplace eigenvalue @math , 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 ( @math ). 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 @math -torus, see (2016) and (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).
We study the asymptotic behaviour of the nodal length of random @math -spherical harmonics @math of high degree @math , i.e. the length of their zero set @math . It is found that the nodal lengths are asymptotically equivalent, in the @math -sense, to the "sample trispectrum", i.e., the integral of @math , the fourth-order Hermite polynomial of the values of @math . A particular by-product of this is a Quantitative Central Limit Theorem (in Wasserstein distance) for the nodal length, in the high energy limit.  Using the multiplicities of the Laplace eigenspace on the sphere (the space of spherical harmonics) we endow the space with Gaussian probability measure. This induces a notion of random Gaussian spherical harmonics of degree n having Laplace eigenvalue E = n(n + 1). We study the length distribution of the nodal lines of random spherical harmonics. Preface 1. Introduction 2. Background results in representation theory 3. Representations of SO(3) and harmonic analysis on S2 4. Background results in probability and graphical methods 5. Spectral representations 6. Characterizations of isotropy 7. Limit theorems for Gaussian subordinated random fields 8. Asymptotics for the sample power spectrum 9. Asymptotics for sample bispectra 10. Spherical needlets and their asymptotic properties 11. Needlets estimation of power spectrum and bispectrum 12. Spin random fields Appendix Bibliography Index.