The link overlap and finite size effects for the 3D Ising spin glass

We study the link overlap between two replicas of an Ising spin glass in th ree dimensions using the Migdal-Kadanoff approximation and scaling argument s based on the droplet picture. For moderate system sizes, the distribution of the link overlap shows the asymmetric shape and large sample-to-sample variations found in Monte-Carlo simulations and usually attributed to repli ca symmetry breaking. However, the scaling of the width of the distribution , and the link overlap in the presence of a weak coupling between the two r eplicas are in agreement with the droplet picture. We also discuss why it i s impossible to see the asymptotic droplet-like behaviour for moderate syst em sizes and temperatures not too far below the critical temperature.