Depinning transition at the upper critical dimension
  We study the effect of quenched random field disorder on a driven elastic interface close to the depinning transition at the upper critical dimension d_{c}=4 using the functional renormalization group. We have found that the displacement correlation function behaves with distance x as ln(x)^{2/3} for large x. Slightly above the depinning transition the force-velocity characteristics is described by the equation v ~ f |ln(f)|^{2/9}, while the correlation length behaves as L_{v} ~ f^{-1/2}|ln(f)|^{1/6}, where f=F/F_{c}-1 is the reduced driving force. 
Andrei A. Fedorenko and Semjon Stepanow