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

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

2006-05-10

arxiv.org/abs/math/0605266v3

Mathematics

Probability

Version 3. Statement of Theorem 3 is corrected

Scientific paper

10.1007/s00220-007-0242-2

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.

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