Fluctuations in the number of nodal domains

Fedor Nazarov, Mikhail Sodin

Research output: Contribution to journalArticlepeer-review


We show that the variance of the number of connected components of the zero set of the two-dimensional Gaussian ensemble of random spherical harmonics of degree n grows as a positive power of n. The proof uses no special properties of spherical harmonics and works for any sufficiently regular ensemble of Gaussian random functions on the two-dimensional sphere with distribution invariant with respect to isometries of the sphere. Our argument connects the fluctuations in the number of nodal lines with those in a random loop ensemble on planar graphs of degree four, which can be viewed as a step toward justification of the Bogomolny-Schmit heuristics.

Original languageEnglish
Article number18588
JournalJournal of Mathematical Physics
Issue number12
StatePublished - 1 Dec 2020


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