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) ≥ 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) ≥ Ct 1/3(log t)−7/3 in the usual sense.
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References
Bernardin C. (2004). Fluctuations in the occupation time of a site in the asymmetric simple exclusion process. Ann. Probab. 32(1B): 855–879
Bertini L. and Giacomin G. (1997). Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183(3): 571–607
Beijeren H., van Kutner R. and Spohn H. (1985). Excess noise for driven diffusive systems. Phys. Rev. Lett. 54: 2026–2029
Bramson M. and Mountford T. (2002). Stationary blocking measures for one-dimensional nonzero mean exclusion processes. Ann. Probab. 30(3): 1082–1130
Ferrari P.A. and Fontes L.R.G. (1994). Current fluctuations for the asymmetric simple exclusion process. Ann. Probab. 22(2): 820–832
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
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
Kardar K., Parisi G. and Zhang Y.Z. (1986). Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56: 889–892
Liggett T.M. (1985). Interacting particle systems. Grundlehren der Mathematischen issenschaften 276. Springer-Verlag, New York
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
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
Landim C. and Yau H.-T. (1997). Fluctuation-dissipation equation of asymmetric simple exclusion processes, Probab. Theory Related Fields 108(3): 321–356
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. 185–204
Sethuraman S. (2003). An equivalence of H −1 norms for the simple exclusion process. Ann. Prob. 31(1): 35–62
Sethuraman S. and Xu L. (1996). A central limit theorem for reversible exclusion and zero-range particle systems. Ann. Probab. 24(4): 1842–1870
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. 3–40
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
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Communicated by H.-T. Yau.
Supported by the Natural Sciences and Engineering Research Council of Canada.
Partially supported by the Hungarian Scientific Research Fund grants T37685 and K60708.
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Quastel, J., Valkó, B. t 1/3 Superdiffusivity of Finite-Range Asymmetric Exclusion Processes on \({\mathbb{Z}}\) . Commun. Math. Phys. 273, 379–394 (2007). https://doi.org/10.1007/s00220-007-0242-2
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DOI: https://doi.org/10.1007/s00220-007-0242-2