On the number of nodal domains of random spherical harmonics

Let $N(f)$ be a number of nodal domains of a random Gaussian spherical harmonic f of degree n. We prove that as n grows to infinity, the mean of N(f)/n² tends to a positive constant a, and that N(f)/n² exponentially concentrates around a. This result is consistent with predictions made by Bogomolny and Schmit using a percolation-like model for nodal domains of random Gaussian plane waves.