Equidistribution of random waves on small balls

Abstract

Following \cite{Ha2} by the first-named author, we continue our investigation of the equidistribution, at small scale, of random Laplacian eigenfunctions on a compact manifold \(\mathbb{M}\). First we generalise the small scale expectation and variance results for random combinations of eigenfunctions to all compact manifolds. Then, assuming the same conditions as in \cite{Ha2}, i.e. 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 improve the equidistribution of random eigenbases in \cite{Ha2} to a smaller scale. In particular, on all \(n\)-\(\dim\) spheres, we prove that the random eigenbasis is almost surely equidistributed up to the scale \(\lambda^{-1/2}\).