Real and complex zeros of Riemannian random waves

Details Real and complex zeros of Riemannian random waves Real and complex zeros of Riemannian random waves

2008-03-30

arxiv.org/abs/0803.4334v1

Mathematics

Spectral Theory

For the Proceedings of the Conference, "Spectral Analysis in Geometry and Number Theory on the occasion of Toshikazu Sunada's

Scientific paper

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.

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