  The reduction of 2D superconducting or antiferromagnetic order by random
dilution is studied as a model for the 2D diluted Heisenberg antiferromagnet
(DHAF) La$_2$Cu$_{1-p}$(Zn,Mg)$_p$O$_4$ and randomly inhomogeneous 2D
suerconductors. We show in simplified models that long-range order can persist
at the percolation threshold despite the presence of disordered one-dimensional
segments, contrary to the classical case. When long-range order persists to the
percolation threshold, charging effects (in the superconductor) or frustrating
interactions (in the antiferromagnet) can dramatically modify the stiffness of
the order. This quantum destruction of stiffness is used to model neutron
scattering data on La$_2$Cu$_{1-p}$(Zn,Mg)$_p$O$_4$. In a certain simplified
model, there is a sharp stiffness transition between ``stiff'' and ``floppy''
ordered phases.
