Real and complex zeros of Riemannian random waves

By Steve Zelditch. 2008

81 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Atmospheric Science, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/12a16e3bc0f9b18b2fec29753c12e81b591de4f4

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In Applied Mathematics

1. Boris Hanin
2. Y. Canzani
3. Jacobus W. Portegies
4. Damien Gayet
5. Jean-Yves Welschinger
6. Steve Zelditch
7. Zhiqin Lu
8. Bernard Shiffman

In Atmospheric Science

1. Boris Hanin
2. Y. Canzani
3. Jacobus W. Portegies
4. Damien Gayet
5. Jean-Yves Welschinger
6. Steve Zelditch

In Geometry and Topology

1. Boris Hanin
2. Y. Canzani
3. Jacobus W. Portegies
4. Damien Gayet
5. Jean-Yves Welschinger
6. Steve Zelditch
7. G. Mikhalkin
8. Eugenii Shustin

Citations in the corpus

• Pluri-potential theory on Grauert tubes of real analytic Riemannian manifolds, I (2011, background): “We gave some initial results on these kernels in [Z2], by somewhat different methods.”
• EXPECTED TOPOLOGY OF RANDOM REAL ALGEBRAIC SUBMANIFOLDS (2013, background): “In a general Riemannian setting, Zelditch proved in [38] the equidistribution of the vanishing locus, whereas critical points of random eigenfunctions of the Laplacian have been addressed by Nicolaescu in [26].”
• EIGENFUNCTIONS AND NODAL SETS (2012, methodology): “We refer to [Z4] for the calculation, which is quite simple because of the invariance under rotations.”
• Scaling Limit for the Kernel of the Spectral Projector and Remainder Estimates in the Pointwise Weyl Law (2014, methodology): “(24) Such a parametrix was used by Zelditch in [41], where a detailed construction was omitted.”
• Embeddings of Riemannian Manifolds with Heat Kernels and Eigenfunctions (2013, background): “DG] [41] Zelditch, S. Fine structure of Zoll spectra.”