Small Scale Equidistribution of Random Eigenbases

Xiaolong Han*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    15 Citations (Scopus)


    We investigate small scale equidistribution of random orthonormal bases of eigenfunctions (i.e., eigenbases) on a compact manifold M. Assume that the group of isometries acts transitively on M and the multiplicity mλ of eigenfrequency λ tends to infinity at least logarithmically as λ → ∞. 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 mλ. In particular, this implies that almost surely random eigenbases on the sphere Sn (n≥ 2) and the tori Tn (n≥ 5) are equidistributed at polynomial scales.

    Original languageEnglish
    Pages (from-to)425-440
    Number of pages16
    JournalCommunications in Mathematical Physics
    Issue number1
    Publication statusPublished - 1 Jan 2017


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