  Recently, based on heuristic arguments, it was conjectured that an intimate
relation exists between any multifractal dimensions, $D_q$ and $D_{q'}$, of the
eigenstates of critical random matrix ensembles: $D_{q'} \approx
qD_q[q'+(q-q')D_q]^{-1}$, $1\le q, q' \le 2$. 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 $\sigma_q$ of the inverse
moments of Wigner delay times, $\bra \tau_{\tbox W}^{-q} \ket \propto
N^{-\sigma_q}$ where $N$ is the linear size of the system, are related to the
level compressibility $\chi$ as $\sigma_q\approx q(1-\chi)[1+q\chi]^{-1}$ for a
limited range of $q$; thus providing a way to probe level correlations by means
of scattering experiments.
