Fluctuations in the number of nodal domains
Probability
2026-05-05 v1 Mathematical Physics
Classical Analysis and ODEs
math.MP
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.
Keywords
Cite
@article{arxiv.2006.07730,
title = {Fluctuations in the number of nodal domains},
author = {Fedor Nazarov and Mikhail Sodin},
journal= {arXiv preprint arXiv:2006.07730},
year = {2026}
}