Multifractal dimensions for critical random matrix ensembles

Based on heuristic arguments we conjecture that an intimate relation exists between the eigenfunction multifractal dimensions $D_q$ of the eigenstates of critical random matrix ensembles $D_{q^{\mathrm{\prime }}}\approx q{D}_q[q^{\mathrm{\prime }}+(q-q^{\mathrm{\prime }})D_q{]}^{-1}$, $1\le q\le 2$. We verify this relation by extensive numerical calculations. We also demonstrate that the level compressibility $\chi$ describing level correlations can be related to $D_q$ in a unified way as $D_q=(1-\chi )[1+(q-1)\chi {]}^{-1}$, thus generalizing existing relations with relevance to the disorder driven Anderson–transition.