Equidistribution of Random Waves on Small Balls

  • Xiaolong Han
  • Melissa TacyEmail author


In this paper, we investigate the small scale equidistribution properties of randomised sums of Laplacian eigenfunctions (i.e. random waves) on a compact manifold. We prove small scale expectation and variance results for random waves on all compact manifolds. Here, “small scale” refers to balls of radius \(r(\lambda )\rightarrow 0\) such that \(r/r_{\text {Planck}}\rightarrow \infty \), where \(r_{\text {Planck}}\) is the Planck scale. For balls at a larger scale (although still \(r(\lambda )\rightarrow 0\)) we also obtain estimates showing that the probability that a random wave fails to equidistribute decays exponentially with the eigenvalue.



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsCalifornia State UniversityNorthridgeUSA
  2. 2.Department of Mathematics and StatisticsUniversity of OtagoDunedinNew Zealand

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