Superdiffusivity of asymmetric exclusion process in dimensions one and   two
  We prove that the diffusion coefficient for the asymmetric exclusion process diverges at least as fast as $t^{1/4}$ in dimension $d=1$ and $(\log t)^{1/2}$ in $d=2$. The method applies to nearest and non-nearest neighbor asymmetric exclusion processes. 
C. Landim, J. Quastel, M. Salmhofer, H.T. Yau