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

philosophical transactions - royal society. mathematical, physical and engineering sciences/philosophical transactions - royal society. mathematical, physical and engineering sciences

Random orthonormal bases of spaces of high dimension