  We investigate the effects of quenched disorder, in the form of site and bond
dilution, on the physics of the $S=1/2$ antiferromagnetic Heisenberg model on
even-leg ladders. Site dilution is found to prune rung singlets and thus create
localized moments which interact via a random, unfrustrated network of
effective couplings, realizing a random-exchange Heisenberg model (REHM) in one
spatial dimension. This system exhibits a power-law diverging correlation
length as the temperature decreases. Contrary to previous claims, we observe
that the scaling exponent is non-universal, i.e., doping dependent. This
finding can be explained by the discrete nature of the values taken by the
effective exchange couplings in the doped ladder. Bond dilution on even-leg
ladders leads to a more complex evolution with doping of correlations, which
are weakly enhanced in 2-leg ladders, and are even suppressed for low dilution
in the case of 4-leg and 6-leg ladders. We clarify the different aspects of
correlation enhancement and suppression due to bond dilution by isolating the
contributions of rung-bond dilution and leg-bond dilution.
