  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.
