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We construct perturbation series for the q-th moment of eigenfunctions ofvarious critical random matrix ensembles in the strong multifractality regimeclose to localization. Contrary to previous investigations, our results arevalid in the region q<1/2. Our findings allow to verify, at first leadingorders in the strong multifractality limit, the symmetry relation for anomalousfractal dimensions Delta(q)=Delta(1-q), recently conjectured for criticalmodels where an analogue of the metal-insulator transition takes place. It isknown that this relation is verified at leading order in the weakmultifractality regime. Our results thus indicate that this symmetry holds inboth limits of small and large coupling constant. For general values of thecoupling constant we present careful numerical verifications of this symmetryrelation for different critical random matrix ensembles. We also present anexample of a system closely related to one of these critical ensembles, butwhere the symmetry relation, at least numerically, is not fulfilled.

Multifractal dimensions for all moments for certain critical random matrix ensembles in the strong multifractality regime