We investigate numerically the inverse participation ratio, $P_2$, of the 3D Anderson model and of the power-law random banded matrix (PRBM) model at criticality. We found that the variance of $\ln P_2$ scales with system size $L$ as ${\sigma }^2\left(L\right)={\sigma }^2\left(\infty \right){-AL}^{{-D}_2/2d}$, with $D_2$ being the correlation dimension and $d$ the system dimension. Therefore the concept of a correlation dimension is well defined in the two models considered. The 3D Anderson transition and the PRBM transition for $b=0.3\mathrm{}$ (see the text for the definition of $b$) are fairly similar with respect to all critical magnitudes studied.

• Received 28 June 2001

DOI:https://doi.org/10.1103/PhysRevLett.88.016401