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Small scale equidistribution of random eigenbases

Authors
  • Han, Xiaolong
Type
Preprint
Publication Date
Nov 03, 2015
Submission Date
Nov 03, 2015
Identifiers
DOI: 10.1007/s00220-016-2597-8
Source
arXiv
License
Yellow
External links

Abstract

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 of eigenfrequency tends to infinity at least logarithmically. 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 multiplicity. In particular, this implies that almost surely random eigenbases on the n-dimensional sphere (n>=2) and the n-dimensional tori (n>=5) are equidistributed at polynomial scales.

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