Small Scale Equidistribution of Random Eigenbases

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 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 S n (n ≥ 2) and the tori T n (n ≥ 5) are equidistributed at polynomial scales.2010 Mathematics Subject Classification. 58G25, 58J65, 35P20, 60B10.

Small scale quantum ergodicity in negatively curved manifolds

“…Berry's random wave conjectures [Berr] suggest that the eigenfunctions of eigenvalue λ 2 behave like random waves with frequency λ. Recent results about equidistribution at various polynomial scales of random waves on manifolds were proved in [Han2,HT,CI]. In comparison, we see that the logarithmical scales in [Han1,HR1] are at much weaker scales.…”

Planck-Scale Mass Equidistribution of Toral Laplace Eigenfunctions

“…There are only a few such results in the literature with r small: -Luo and Sarnak [16] showed this for r > E −α for some small α > 0, for the modular surface, where the eigenfunctions are the eigenfunctions of all Hecke operators, and Young [18] showed this for all eigenfunctions for r > E −1/4+o (1) , assuming the Generalized Riemann Hypothesis; -Hezari-Rivière [13] and Han [11] showed the integral is the expected value up to a multiplicative constant, for r > (log E) −α for some small α > 0, on negatively curved manifolds. Han [12] also showed this for "symmetric" manifolds (i.e. manifolds on which the group of isometries act transitively) on which the lower bound on r depends on the growth rate of the eigenspace dimensions (the "spectral degeneracy").…”

Small Scale Equidistribution for a Point Scatterer on the Torus

“…Hezari and Rivière [12] and Han [8] established (a non-uniform version of) (1.1) with balls A = B r (x) of radii r > (log E j k ) −α on compact negatively curved manifolds. Further results are due to Han [9] (small scale equidistribution for random eigenbases on a certain class of "symmetric" manifolds), Han and Tacy [10] (random combinations of Laplace eigenfunctions on compact manifolds), Humphries [14] (small scale equidistribution for Hecke-Maass forms, with balls A = B r (x) whose centres are random. See also [7,28] for results on the torus), and de Courcy-Ireland [6] (discrepancy estimates for random spherical harmonics).…”