Random orthonormal bases of spaces of high dimension

Abstract

We consider a sequence H_N of Hilbert spaces of dimensions d_N tending to infinity. The motivating examples are eigenspaces or quasi-mode spaces of a Laplace or Schrodinger operator. We define a random ONB of H_N by fixing one ONB and changing it by a random element of U(d_N). A random ONB of the direct sum of the H_N is an independent sequence {U_N} of random ONB's of the H_N. We prove that if d_N tends to infinity and if the normalized traces of observables in H_N tend to a unique limit state, then a random ONB also tends to that limit state. This generalizes an earlier result of the author for eigenspaces of the standard 2-sphere, and shows that the result does not depend on how fast the dimensions grow. In particular it is valid for eigenspaces of a flat rational torus in dimensions > 4. The main idea is to convert quantum ergodicity into a problem on the moments of inertia of permutahedra and to calculate the moments using Schur polynomials.