We consider finite-range asymmetric exclusion processes on $\mathbb Z$ with non-zero drift. The diffusivity $D(t)$ is expected to be of ${ \mathcal O}(t^{ 1/3})$. We prove that $D(t)\ge Ct^{ 1/3}$ in the weak (Tauberian) sense that $\int_0^\infty e^{ -\lambda t}tD(t)dt \ge C\lambda^{ -7/3}$ as $\lambda\to 0$. The proof employs the resolvent method to make a direct comparison with the totally asymmetric simple exclusion process, for which the result is a consequence of the scaling limit for the two-point function recently obtained by Ferrari and Spohn. In the nearest neighbor case, we show further that $tD(t)$ is monotone, and hence we can conclude that $D(t)\ge Ct^{ 1/3}(\log t)^{ -7/3}$ in the usual sense.
Categories: math.PR math-ph math.MP