Fluctuations in the number of nodal domains

TL;DR

This paper demonstrates that the variance of the number of nodal domains in Gaussian spherical harmonics grows with degree n, linking fluctuations to random loop ensembles and supporting Bogomolny-Schmit heuristics.

Contribution

It establishes a general growth result for the variance of nodal domains in Gaussian ensembles, independent of specific harmonic properties.

Findings

Variance grows as a positive power of n

Connects nodal line fluctuations to planar graph loop ensembles

Supports Bogomolny-Schmit heuristics

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.