Communications in Mathematical Physics

, Volume 273, Issue 2, pp 379–394

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

Article

Abstract

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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References

  1. B.
    Bernardin C. (2004). Fluctuations in the occupation time of a site in the asymmetric simple exclusion process. Ann. Probab. 32(1B): 855–879 MATHCrossRefMathSciNetGoogle Scholar
  2. BG.
    Bertini L. and Giacomin G. (1997). Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183(3): 571–607 MATHCrossRefADSMathSciNetGoogle Scholar
  3. BKS.
    Beijeren H., van Kutner R. and Spohn H. (1985). Excess noise for driven diffusive systems. Phys. Rev. Lett. 54: 2026–2029 CrossRefADSMathSciNetGoogle Scholar
  4. BM.
    Bramson M. and Mountford T. (2002). Stationary blocking measures for one-dimensional nonzero mean exclusion processes. Ann. Probab. 30(3): 1082–1130 MATHCrossRefMathSciNetGoogle Scholar
  5. FF.
    Ferrari P.A. and Fontes L.R.G. (1994). Current fluctuations for the asymmetric simple exclusion process. Ann. Probab. 22(2): 820–832 MATHMathSciNetGoogle Scholar
  6. FS.
    Ferrari P.L. and Spohn H. (2006). Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Commun. Math. Phys. 265(1): 1–44 MATHCrossRefADSMathSciNetGoogle Scholar
  7. FNS.
    Forster D., Nelson D. and Stephen M.J. (1977). Large-distance and long time properties of a randomly stirred fluid. Phys. Rev. A 16: 732–749 CrossRefADSMathSciNetGoogle Scholar
  8. KPZ.
    Kardar K., Parisi G. and Zhang Y.Z. (1986). Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56: 889–892 MATHCrossRefADSGoogle Scholar
  9. L.
    Liggett T.M. (1985). Interacting particle systems. Grundlehren der Mathematischen issenschaften 276. Springer-Verlag, New York Google Scholar
  10. LOY.
    Landim C., Olla S. and Yau H.T. (1996). Some properties of the diffusion coefficient for asymmetric simple exclusion processes. Ann. Probab. 24(4): 1779–1808 MATHCrossRefMathSciNetGoogle Scholar
  11. LQSY.
    Landim C., Quastel J., Salmhofer M. and Yau H.-T. (2004). Superdiffusivity of asymmetric exclusion process in dimensions one and two. Commun. Math. Phys. 244(3): 455–481 MATHCrossRefADSMathSciNetGoogle Scholar
  12. LY.
    Landim C. and Yau H.-T. (1997). Fluctuation-dissipation equation of asymmetric simple exclusion processes, Probab. Theory Related Fields 108(3): 321–356 MATHCrossRefMathSciNetGoogle Scholar
  13. PS.
    Prähofer, M., Spohn, H.: Current fluctuations for the totally asymmetric simple exclusion process. In: In and out of equilibrium (Mambucaba, 2000), Progr. Probab. 51, Boston, MA: Birkhäuser Boston, 2002, pp. 185–204Google Scholar
  14. S.
    Sethuraman S. (2003). An equivalence of H −1 norms for the simple exclusion process. Ann. Prob. 31(1): 35–62 MATHCrossRefMathSciNetGoogle Scholar
  15. SX.
    Sethuraman S. and Xu L. (1996). A central limit theorem for reversible exclusion and zero-range particle systems. Ann. Probab. 24(4): 1842–1870 MATHCrossRefMathSciNetGoogle Scholar
  16. V.
    Varadhan, S.R.S.: Lectures on hydrodynamic scaling. In: Hydrodynamic limits and related topics (Toronto, ON, 1998), Fields Inst. Commun. 27, Providence, RI: Amer. Math. Soc., 2000, pp. 3–40Google Scholar
  17. Y.
    Yau H.-T. (2004). (log t)2/3 law of the two dimensional asymmetric simple exclusion process. Ann. of Math. (2) 159(1): 377–405 MATHMathSciNetCrossRefGoogle Scholar

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© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Departments of Mathematics and StatisticsUniversity of TorontoTorontoCanada

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