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 $D_{q'} \approx qD_q[q'+(q-q')D_q]^{-1}$, $1\le q \le 2$. We verify this relation by extensive numerical calculations. We also demonstrate that the level compressibility $χ$ describing level correlations can be related to $D_q$ in a unified way as $D_q=(1-χ)[1+(q-1)χ]^{-1}$, thus generalizing existing relations with relevance to the disorder driven Anderson--transition.

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