×

zbMATH — the first resource for mathematics

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.

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[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 · doi:10.1214/009117905000000233
[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 · doi:10.1023/A:1018615306992
[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) · doi:10.1103/PhysRevLett.54.2026
[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) · doi:10.1103/PhysRevE.65.017105
[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 · doi:10.1007/s00220-004-1204-6
[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) · doi:10.1103/PhysRevA.16.732
[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 · doi:10.1016/j.nuclphysb.2004.07.030
[10] Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000) · Zbl 0969.15008 · doi:10.1007/s002200050027
[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 · doi:10.1103/PhysRevLett.56.889
[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) · doi:10.1103/PhysRevA.45.638
[14] Liggett, T.M.: Coupling the simple exclusion process. Ann. Probab. 4, 339–356 (1976) · Zbl 0339.60091 · doi:10.1214/aop/1176996084
[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 · doi:10.1007/PL00001398
[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 · doi:10.1023/A:1019791415147
[21] Prähofer, M., Spohn, H.: Exact scaling function for one-dimensional stationary KPZ growth. J. Stat. Phys. 115, 255–279 (2004) · Zbl 1157.82363 · doi:10.1023/B:JOSS.0000019810.21828.fc
[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) · doi:10.1007/BF01318418
[24] Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994) · Zbl 0789.35152 · doi:10.1007/BF02100489
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.