Multifractal dimensions for critical random matrix ensembles

Based on heuristic arguments we conjecture that an intimate relation exists between the eigenfunction multifractal dimensions Dq of the eigenstates of critical random matrix ensembles Dq′≈qDq[q′ +(q− q′ )Dq]−1, 1⩽q⩽2. We verify this relation by extensive numerical calculations. We also demonstrate that the level compressibility χ describing level correlations can be related to Dq in a unified way as Dq=(1− χ)[1+(q−1)χ]−1, thus generalizing existing relations with relevance to the disorder-driven Anderson transition