$t^{1\mathrm{/}3}$ Superdiffusivity of Finite-Range Asymmetric Exclusion Processes on $\mathbb{Z}$

We consider finite-range asymmetric exclusion processes on $\mathbb{Z}$ with non-zero drift. The diffusivity $D(t)$ is expected to be of ${\mathcal O}(t^{1/3})$\mathrm{.}Weprovethat$D(t)\ge Ct^{1/3}$ in the weak (Tauberian) sense that ${\int }_0^{\mathrm{\infty }}{e}^{-\lambda t}tD(t)dt\ge C{\lambda }^{-7\mathrm{/}3}$ 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.