  We study long-range power-law correlated disorder on square and cubic
lattices. In particular, we present high-precision results for the percolation
thresholds and the fractal dimension of the largest clusters as function of the
correlation strength. The correlations are generated using a discrete version
of the Fourier filtering method. We consider two different metrics to set the
length scales over which the correlations decay, showing that the percolation
thresholds are highly sensitive to such system details. By contrast, we verify
that the fractal dimension $d_{\rm f}$ is a universal quantity and unaffected
by the choice of metric. We also show that for weak correlations, its value
coincides with that for the uncorrelated system. In two dimensions we observe a
clear increase of the fractal dimension with increasing correlation strength,
approaching $d_{\rm f}\rightarrow 2$. The onset of this change does not seem to
be determined by the extended Harris criterion.
