Sign-balance of random Laplace eigenfunctions

TL;DR

This paper investigates the sign distribution of Laplace eigenfunctions, proving that random eigenfunctions are sign-balanced above an optimal scale, with implications for deterministic eigenfunctions and applications to spherical harmonics.

Contribution

It introduces a strong notion of sign-balance, proves random eigenfunctions are sign-balanced at an optimal scale, and extends this to arbitrary levels on Riemannian manifolds.

Findings

Random eigenfunctions are sign-balanced above a specific scale.

The optimal scale for sign-balance is determined up to a logarithmic factor.

Results include random spherical harmonics and band-limited random waves.

Abstract

Motivated by the problem of the small-scale sign distribution of Laplace eigenfunctions, we introduce a strong notion of sign-balance for (eigen)functions, and prove that random eigenfunctions are sign-balanced above a precisely determined scale with almost full probability. The scale is proven to be optimal up to a logarithmic power of the energy. Our results include the important case of random spherical harmonics, as well as more general band-limited random waves on smooth Riemannian manifolds. Extending the notion of balance to arbitrary levels, we determine the precise optimum scale above which random eigenfunctions are volume-balanced with respect to non-zero levels. Beyond their intrinsic interest, our results serve as a model for a natural conjecture on the optimal scale at which deterministic Laplace eigenfunctions are sign-balanced.