Fluctuations in the number of nodal domains
- 1 December 2020
- journal article
- research article
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 61 (12), 123302
- https://doi.org/10.1063/5.0018588
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 toward justification of the Bogomolny–Schmit heuristics.Keywords
Funding Information
- Division of Mathematical Sciences (1900008)
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