Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. (English) Zbl 1118.82032

The totally asymmetric simple exclusion process (TASEP) on the one-dimensional lattice with the Bernoulli \(\rho\) measure as initial conditions, \(0< \rho <1\), is stationary in space and time. Let \(N_t(j)\) be the number of particles which have crossed the bond from \(j\) to \(j+1\) during the time span \([0,t]\). For \( j = (1-2p)t + 2w(\rho(1-\rho))^{1/3}t^{2/3}\) the authors prove that the fluctuations of \(N_t(j)\) for large \(t\) are of order \(t^{1/3}\) and determine the limiting distribution function \(F_w(s)\), which is a generalization of the GUE (Gaussian unitary ensemble) Tracy-Widom distribution. The family \(F_w(s)\) of distribution functions have been obtained before by J. Baik and E. M. Rains [J. Stat. Phys. 100, 523–541 (2000; Zbl 0976.82043)] in the context of the PNG model (polynuclear growth model) with boundary sources, which requires the asymptotics of a Riemann-Hilbert problem. In this work the authors arrive at \(F_w(s)\) through the asymptotics of a Fredholm determinant. \(F_w(s)\) is simply related to the scaling function for the space-time covariance of the stationary TASEP, equivalently to the asymptotic transition probability of a single second class particle.


82C22 Interacting particle systems in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics


Zbl 0976.82043
Full Text: DOI arXiv


[1] Baik, J., Ben Arous, G., Péché, S.: Phase transition of the largest eigenvalue for non-null complex sample covariance matrices. Ann. Probab. 33, 1643–1697 (2005) · Zbl 1086.15022
[2] Baik, J., Rains, E.M.: Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 100, 523–542 (2000) · Zbl 0976.82043
[3] Baik, J., Rains, E.M.: Symmetrized random permutations. In: Random Matrix Models and Their Applications, Vol. 40, pp. 1–19. Cambridge University Press, Cambridge (2001) · Zbl 0989.60010
[4] van Beijeren, H., Kutner, R., Spohn, H.: Excess noise for driven diffusive systems. Phys. Rev. Lett. 54, 2026–2029 (1985)
[5] Colaiori, F., Moore, M.A.: Numerical solution of the mode-coupling equations for the Kardar-Parisi-Zhang equation in one dimension. Phys. Rev. E 65, 017105 (2002)
[6] Ferrari, P.L.: Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues. Commun. Math. Phys. 252, 77–109 (2004) · Zbl 1124.82316
[7] Ferrari, P.L., Spohn, H.: A determinantal formula for the GOE Tracy-Widom distribution. J. Phys. A 38, L557–L561 (2005)
[8] Forster, D., Nelson, D.R., Stephen, M.J.: Large-distance and long-time properties of a randomly stirred fluid. Phys. Rev. A 16, 732–749 (1977)
[9] Imamura, T., Sasamoto, T.: Fluctuations of the one-dimensional polynuclear growth model with external sources. Nucl. Phys. B 699, 503–544 (2004) · Zbl 1123.82352
[10] Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000) · Zbl 0969.15008
[11] Johansson, K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242, 277–329 (2003) · Zbl 1031.60084
[12] Kardar, K., Parisi, G., Zhang, Y.Z.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986) · Zbl 1101.82329
[13] Krug, J., Meakin, P., Halpin-Healy, T.: Amplitude universality for driven interfaces and directed polymers in random media. Phys. Rev. A 45, 638–653 (1992)
[14] Liggett, T.M.: Coupling the simple exclusion process. Ann. Probab. 4, 339–356 (1976) · Zbl 0339.60091
[15] Liggett, T.M.: Stochastic interacting systems: contact, voter and exclusion processes. Springer Verlag, Berlin (1985) · Zbl 0949.60006
[16] Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and theorems for the special functions of mathematical physics. Grundlehren Band 52, Springer Verlag, Berlin (1966) · Zbl 0143.08502
[17] Okounkov, A.: Infinite wedge and random partitions. Selecta Math. 7, 57–81 (2001) · Zbl 0986.05102
[18] Prähofer, M.: Stochastic surface growth. Ph.D. thesis, Ludwig-Maximilians-Universität, München. Available at: http://edoc.ub.uni-muenchen.de/archive/00001381, 2003
[19] Prähofer, M., Spohn, H.: Current fluctuations for the totally asymmetric simple exclusion process. In: In and out of equilibrium (V. Sidoravicius, ed.), Progress in Probability, Boston Basel: Birkhäuser, 2002 · Zbl 1015.60093
[20] Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1071–1106 (2002) · Zbl 1025.82010
[21] Prähofer, M., Spohn, H.: Exact scaling function for one-dimensional stationary KPZ growth. J. Stat. Phys. 115, 255–279 (2004) · Zbl 1157.82363
[22] Sasamoto, T.: Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38, L549–L556 (2005)
[23] Spohn, H.: Excess noise for a lattice gas model of a resistor. Z. Phys. B 57, 255–261 (1984)
[24] Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994) · Zbl 0789.35152
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