Fluctuations of the inverse participation ratio at the Anderson transition

Statistics of the inverse participation ratio (IPR) at the critical point o f the localization transition is studied numerically for the power-law rand om banded matrix model. It is shown that the IPR distribution function is s cale invariant, with a power-law asymptotic "tail." This scale invariance i mplies that the fractal dimensions D-q are nonfluctuating quantities, contr ary to a recent claim in the literature. A recently proposed relation betwe en D-2 and the spectral compressibility chi is violated in the regime of st rong multifractality, with chi --> 1 in the limit D-2 --> 0.