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Article Dans Une Revue Bernoulli Année : 2023

Almost sure asymptotics for Riemannian random waves

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Louis Gass
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Résumé

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 band and monochro-matic 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 is reminiscent of Berry's conjecture and 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, these asymp-totics 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 lines.

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Dates et versions

hal-02867553 , version 1 (14-06-2020)

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Louis Gass. Almost sure asymptotics for Riemannian random waves. Bernoulli, 2023, 29 (1), pp.625-651. ⟨10.3150/22-BEJ1471⟩. ⟨hal-02867553⟩
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