On the generalized dimensions of multifractal eigenstates

Recently, based on heuristic arguments, it was conjectured that an intimate relation exists between any multifractal dimensions, Dq and , of the eigenstates of critical random matrix ensembles: , 1 ⩽ q, q′ ⩽ 2. Here, we verify this relation by extensive numerical calculations on critical random matrix ensembles and extend its applicability to q < 1/2, but also to deterministic models producing multifractal eigenstates and to generic multifractal structures. We also demonstrate, for the scattering version of the power-law banded random matrix model at criticality, that the scaling exponents σq of the inverse moments of Wigner delay times, where N is the linear size of the system, are related to the level compressibility χ as σq ≈ q (1 − χ)[1 + qχ]−1 for a limited range of q, thus providing a way to probe level correlations by means of scattering experiments.