We study the non-equilibrium behaviour of three-dimensional spin glasses in the Migdal-Kadanoff approximation. This approximation is exact for disordered hierarchical lattices which have a unique ground state and equilibrium properties correctly described by the droplet model. Extensive numerical simulations show that this model lacks ageing in the remanent magnetization as well as a maximum in the magnetic viscosity in disagreement with experiments as well as with numerical studies of the Edwards-Anderson model. This result strongly limits the validity of the droplet model (at least in its simplest form) as a good model for real spin glasses.
Recently, a method has been proposed to obtain accurate predictions for low-temperature properties of lattice spin glasses that is practical even above the upper critical dimension,
dc=6. This method is based on the observation that bond-dilution enables the numerical treatment of larger lattices, and that the subsequent combination of such data at various bond densities into a finite-size scaling ansatz produces more robust scaling behavior. In the present study we test the potential of such a procedure, in particular, to obtain the stiffness exponent for the hierarchical Migdal-Kadanoff lattice. Critical exponents for this model are known with great accuracy and any simulations can be executed to very large lattice sizes at almost any bond density, effecting an insightful comparison that highlights the advantages---as well as the weaknesses---of this method. These insights are applied to the Edwards-Anderson model in
d=3 with Gaussian bonds.