Let f be a spherical harmonics of degree n on the 2-dimensional sphere, and let N(f) be the number of connected components of the zero set {f=0}. The quantity N(f) can be viewed as a topological complexity of the nodal picture of f. The celebrated Courant nodal domain theorem yields that N(f) cannot be much larger than the square of n.
Suppose f is a random spherical harmonics. What can be said about N(f) as n tends to infinity? For instance, whether the expectation of N(f) is comparable with the square of n, or it has a more slow growth? Several years ago, some non-rigorous predictions based on a percolation-like model were made by Bogomolny and Schmit. We prove that the expectation of N(f) indeed grows like the square of n with exponential concentration around its mean.
(joint work with Fedor Nazarov)
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