Real and complex zeros of Riemannian random waves

TL;DR

This paper studies the distribution of real and complex zeros of Riemannian random waves, showing uniform distribution of real zeros and convergence of complex zeros to a limit current on the complexified manifold.

Contribution

It establishes the asymptotic distribution of zeros of Riemannian random waves in both real and complex settings, extending understanding of eigenfunction behavior.

Findings

Real zeros are uniformly distributed with respect to the volume form.

Complex zeros of analytic continuations tend to a limit current.

Results apply to Gaussian random linear combinations of Laplacian eigenfunctions.

Abstract

We consider Riemannian random waves, i.e. Gaussian random linear combination of eigenfunctions of the Laplacian on a compact Riemannian manifold with frequencies from a short interval (`asymptotically fixed frequency'). We first show that the expected limit distribution of the real zero set of a is uniform with respect to the volume form of a compact Riemannian manifold $(M, g)$. We then show that the complex zero set of the analytic continuations of such Riemannian random waves to a Grauert tube in the complexification of $M$ tends to a limit current.