Real and complex zeros of Riemannian random waves
Spectral Theory
2012-07-11 v1 Probability
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 . 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 tends to a limit current.
Cite
@article{arxiv.0803.4334,
title = {Real and complex zeros of Riemannian random waves},
author = {Steve Zelditch},
journal= {arXiv preprint arXiv:0803.4334},
year = {2012}
}
Comments
For the Proceedings of the Conference, "Spectral Analysis in Geometry and Number Theory on the occasion of Toshikazu Sunada's 60th birthday", to appear in the Contemp. Math. Series