  The two-dimensional antiferromagnetic S=1/2 Heisenberg model with random site
dilution is studied using quantum Monte Carlo simulations. Ground state
properties of the largest connected cluster on L*L lattices, with L up to 64,
are calculated at the classical percolation threshold. In addition, clusters
with a fixed number Nc of spins on an infinite lattice at the percolation
density are studied for Nc up to 1024. The disorder averaged sublattice
magnetization per spin extrapolates to the same non-zero infinite-size value
for both types of clusters. Hence, the percolating clusters, which are fractal
with dimensionality d=91/48, have antiferromagnetic long-range order. This
implies that the order-disorder transition driven by site dilution occurs
exactly at the percolation threshold and that the exponents are classical. The
same conclusion is reached for the bond-diluted system. The full sublattice
magnetization versus site-dilution curve is obtained in terms of a
decomposition into a classical geometrical factor and a factor containing all
the effects of quantum fluctuations. The spin stiffness is shown to obey the
same scaling as the conductivity of a random resistor network.
