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We discuss the localization behavior of localized electronic wave functionsin the one- and two-dimensional tight-binding Anderson model with diagonaldisorder. We find that the distributions of the local wave function amplitudesat fixed distances from the localization center are well approximated bylog-normal fits which become exact at large distances. These fits areconsistent with the standard single parameter scaling theory for the Andersonmodel in 1d, but they suggest that a second parameter is required to describethe scaling behavior of the amplitude fluctuations in 2d. From the log-normaldistributions we calculate analytically the decay of the mean wave functions.For short distances from the localization center we find stretched exponentiallocalization ("sublocalization") in both, 1d and 2d. In 1d, for largedistances, the mean wave functions depend on the number of configurations Nused in the averaging procedure and decay faster that exponentially("superlocalization") converging to simple exponential behavior only in theasymptotic limit. In 2d, in contrast, the localization length increaseslogarithmically with the distance from the localization center andsublocalization occurs also in the second regime. The N-dependence of the meanwave functions is weak. The analytical result agrees remarkably well with thenumerical calculations.Comment: 12 pages with 9 figures and 1 tabl

Sublocalization, superlocalization, and violation of standard single-parameter scaling in the Anderson model