Multifractality and critical fluctuations at the Anderson transition

Critical fluctuations of wave functions and energy levels at the Anderson transition are studied for the family of the power-law random banded matrix ensembles. It is shown that the distribution functions of the inverse participation ratios (IPR) $P_q$ are scale-invariant at the critical point, with a power-law asymptotic tail. The IPR distribution, the multifractal spectrum, and the level statistics are calculated analytically in the limits of weak and strong couplings, as well as numerically in the full range of couplings.