On the generalized dimensions of multifractal eigenstates

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 $σ_q$ of the inverse moments of Wigner delay times, $\bra τ_{\tbox W}^{-q} \ket \propto N^{-σ_q}$ where $N$ is the linear size of the system, are related to the level compressibility $χ$ as $σ_q\approx q(1-χ)[1+qχ]^{-1}$ for a limited range of $q$; thus providing a way to probe level correlations by means of scattering experiments.

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