Small Scale Equidistribution of Random Eigenbases

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 m_λ 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 m_λ. 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.