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arXiv · 2006.07730

Fluctuations in the number of nodal domains

Fedor Nazarov·Mikhail Sodin
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Abstract

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 towards justification of the Bogomolny-Schmit heuristics.

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Fedor Nazarov, Mikhail Sodin. 2020-06-13. Fluctuations in the number of nodal domains. https://doi.org/10.1063/5.0018588

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