On the number of nodal domains of random spherical harmonics

Abstract

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 $a$, and that $N(f)/n^2$ 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.