Disordered periodic systems at the upper critical dimension

The effects of weak pointlike disorder on periodic systems at their upper c ritical dimension D-c for disorder are studied. The systems studied range f rom simple elastic systems with D (c) = 4 to systems with long-range intera ctions with D-c = 2 and systems with D-c = 3 such as the vortex lattice wit h dispersive elastic constants. These problems are studied using the Gaussi an variational method and the functional renormalization group. In all the cases studied we find a typical ultraslow ln ln(x) growth of the asymptotic displacement correlation function, resulting in nearly perfect translation al order. Consequences for the Bragg glass phase of vortex lattices are dis cussed.