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Small Scale Equidistribution of Random Eigenbases

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Abstract

We investigate small scale equidistribution of random orthonormal bases of eigenfunctions (i.e., eigenbases) on a compact manifold \({{\mathbb M}}\). Assume that the group of isometries acts transitively on \({{\mathbb M}}\) and the multiplicity \({m_\lambda}\) of eigenfrequency \({\lambda}\) tends to infinity at least logarithmically as \({\lambda \to \infty}\). We prove that, with respect to the natural probability measure on the space of eigenbases, almost surely a random eigenbasis is equidistributed at small scales; furthermore, the scales depend on the growth rate of \({m_\lambda}\). In particular, this implies that almost surely random eigenbases on the sphere \({{\mathbb S}^n}\) (\({n \ge 2}\)) and the tori \({{\mathbb T}^n}\) (\({n \ge 5}\)) are equidistributed at polynomial scales.

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Authors and Affiliations

  1. Department of Mathematics, The Australian National University, Canberra, ACT, 2601, Australia

    Xiaolong Han

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  1. Xiaolong Han
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Correspondence to Xiaolong Han.

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Communicated by S. Zelditch

Research is partially supported by the Australian Research Council through Discovery Projects DP120102019 and DP150102419.

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Han, X. Small Scale Equidistribution of Random Eigenbases. Commun. Math. Phys. 349, 425–440 (2017). https://doi.org/10.1007/s00220-016-2597-8

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  • DOI: https://doi.org/10.1007/s00220-016-2597-8

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