Universal and non-universal features of the multifractality exponents of critical wavefunctions

We calculate perturbatively the multifractality spectrum of wavefunctions in critical random matrix ensembles in the regime of weak multifractality. We show that in the leading order the spectrum is universal, while the higher order corrections are model-specific. Explicit results for the anomalous dimensions are derived in the power-law and ultrametric random matrix ensembles.