Abstract
We consider the Riemannian random wave model of Gaussian linear combinations of Laplace eigenfunctions on a general compact Riemannian manifold. With probability one with respect to the Gaussian coefficients, we establish that, both for large and small band models, the process properly rescaled and evaluated at an independently and uniformly chosen point X on the manifold, converges in distribution under the sole randomness of X towards an universal Gaussian field as the frequency tends to infinity. This result extends the celebrated central limit Theorem of Salem–Zygmund for trigonometric polynomials series to the more general framework of compact Riemannian manifolds. We then deduce from the above convergence the almost-sure asymptotics of the nodal volume associated with the random wave. To the best of our knowledge, in the real Riemannian case, these asymptotics were only known in expectation and not in the almost sure sense due to the lack of sufficiently accurate variance estimates. This in particular addresses a question of S. Zelditch regarding the almost sure equidistribution of nodal volume.
Acknowledgements
The author is grateful to Jürgen Angst and Guillaume Poly for suggesting the idea of this paper and supporting him during the writing process. The author also thanks Thomas Letendre for a very careful reading of the paper, spotting many typos and giving relevant suggestions to improve the reading of the paper. Finally, the author thank the anonymous referee for his careful reading of the paper.
Citation
Louis Gass. "Almost-sure asymptotics for Riemannian random waves." Bernoulli 29 (1) 625 - 651, February 2023. https://doi.org/10.3150/22-BEJ1471
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