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)/n^2 tends to a positive constant, and that N(f)/n^2 exponentially concentrates around that constant. This result is consistent with predictions made by Bogomolny and Schmit using a percolation-like model for nodal domains of random Gaussian plane waves.