  We have successfully grown sizable single crystals of
La_2Cu_{1-z}(Zn,Mg)_zO_4 with up to nearly half of the magnetic Cu sites
replaced by non-magnetic Zn and Mg. Neutron scattering, SQUID magnetometry, and
complementary quantum Monte Carlo (QMC) simulations demonstrate that this
material is an excellent model system for the study of site percolation of the
square-lattice Heisenberg antiferromagnet (SLHAF) in the quantum-spin limit
S=1/2. Carefully oxygen-reduced samples exhibit Neel order up to the
percolation threshold for site dilution, z_p ~ 40.7%. Up to at least z = 35%,
the Neel temperature T_N(z) of the experimental system corresponds to the
temperature at which QMC indicates that the spin correlations for the
nearest-neighbor S=1/2 SLHAF have grown to approximately 100 lattice constants.
Neutron scattering measurements of the static structure factor in the
paramagnetic regime allow the determination of the two-dimensional spin
correlations, which are found to be in excellent quantitative agreement with
QMC over a wide common temperature and doping range. Neutron scattering and QMC
results for the temperature dependence of the static structure factor amplitude
S(\pi,\pi) are in good agreement as well. The combined experimental and
numerical data presented here provide valuable quantitative information for
tests of theories of the randomly-diluted S=1/2 SLHAF.
