Communications in Mathematical Physics

, Volume 273, Issue 2, pp 379–394

t1/3 Superdiffusivity of Finite-Range Asymmetric Exclusion Processes on \({\mathbb{Z}}\)


    • Departments of Mathematics and StatisticsUniversity of Toronto
  • Benedek Valkó
    • Departments of Mathematics and StatisticsUniversity of Toronto

DOI: 10.1007/s00220-007-0242-2

Cite this article as:
Quastel, J. & Valkó, B. Commun. Math. Phys. (2007) 273: 379. doi: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) ≥ Ct1/3 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) ≥ Ct1/3(logt)−7/3 in the usual sense.

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© Springer-Verlag 2007