Fluctuation properties of the TASEP with periodic initial configuration. (English) Zbl 1136.82028

Summary: We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process. We then consider certain periodic initial conditions and determine the kernel in the scaling limit. This result has been announced first in a letter by one of us [T. Sasamoto, Spatial correlations of the 1D KPZ surface on a flat substrate, J. Phys. A 38, L549–L556 (2005), doi:10.1088/0305-4470/38/33/L01)] and here we provide a self-contained derivation. Connections to last passage directed percolation and random matrices are also briefly discussed.


82C22 Interacting particle systems in time-dependent statistical mechanics
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
82C43 Time-dependent percolation in statistical mechanics
15B52 Random matrices (algebraic aspects)
Full Text: DOI arXiv Link


[1] 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
[2] Barabási, A.L., Stanley, H.E.: Fractal Concepts in Surface Growth. Cambridge University Press, Cambridge (1995) · Zbl 0838.58023
[3] Borodin, A., Rains, E.M.: Eynard-Mehta theorem, Schur process, and their Pfaffian analogs. J. Stat. Phys. 121, 291–317 (2005) · Zbl 1127.82017
[4] Borodin, A., Ferrari, P.L., Prähofer, M.: Fluctuations in the discrete TASEP with periodic initial configurations and the Airy1 process. Int. Math. Res. Pap. rpm002 (2007) · Zbl 1136.82321
[5] Deift, P.: Universality for mathematical and physical systems. arXiv:math-ph/0603038 (2006)
[6] Dyson, F.J.: A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191–1198 (1962) · Zbl 0111.32703
[7] Eynard, B., Mehta, M.L.: Matrices coupled in a chain I. Eigenvalue correlations. J. Phys. A 31, 4449–4456 (1998) · Zbl 0938.15012
[8] 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
[9] Ferrari, P.L.: Shape fluctuations of crystal facets and surface growth in one dimension. Ph.D. thesis, Technische Universität München. http://tumb1.ub.tum.de/publ/diss/ma/2004/ferrari.html (2004)
[10] Ferrari, P.L., Prähofer, M.: One-dimensional stochastic growth and Gaussian ensembles of random matrices. Markov Process. Relat. Fields 12, 203–234 (2006) · Zbl 1136.82335
[11] Ferrari, P.L., Spohn, H.: A determinantal formula for the GOE Tracy-Widom distribution. J. Phys. A 38, L557–L561 (2005)
[12] Ferrari, P.L., Spohn, H.: Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Commun. Math. Phys. 265, 1–44 (2006) · Zbl 1118.82032
[13] Forrester, P.J., Nagao, T., Honner, G.: Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges. Nucl. Phys. B 553, 601–643 (1999) · Zbl 0944.82012
[14] Hough, J.B., Krishnapur, M., Peres, Y., Virag, B.: Determinantal processes and independence. Probab. Surv. 3, 206–229 (2006) · Zbl 1189.60101
[15] Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000) · Zbl 0969.15008
[16] Johansson, K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242, 277–329 (2003) · Zbl 1031.60084
[17] Johansson, K.: Random matrices and determinantal processes. In: Bovier, A., Dunlop, F., van Enter, A., den Hollander, F., Dalibard, J. (eds.) Mathematical Statistical Physics. Lecture Notes of the Les Houches Summer School 2005, vol. LXXXIII, pp. 1–56. Elsevier, Amsterdam (2006) · Zbl 1411.60144
[18] Kardar, K., Parisi, G., Zhang, Y.Z.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986) · Zbl 1101.82329
[19] Karlin, S., McGregor, L.: Coincidence probabilities. Pac. J. 9, 1141–1164 (1959) · Zbl 0092.34503
[20] Koekoek, R., Swarttouw, R.F.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. arXiv:math.CA/9602214 (1996)
[21] Liggett, T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin (1999) · Zbl 0949.60006
[22] Lyons, R.: Determinantal probability measures. Publ. Math. Inst. Hautes Etudes Sci. 98, 167–212 (2003) · Zbl 1055.60003
[23] Meakin, P.: Fractals, Scaling and Growth Far from Equilibrium. Cambridge University Press, Cambridge (1998) · Zbl 1064.37500
[24] Nagao, T., Sasamoto, T.: Asymmetric simple exclusion process and modified random matrix ensembles. Nucl. Phys. B 699, 487–502 (2004) · Zbl 1123.82345
[25] Okounkov, A., Reshetikhin, N.: Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Am. Math. Soc. 16, 581–603 (2003) · Zbl 1009.05134
[26] Øksendal, B.K.: Stochastic Differential Equations, 5th ed. Springer, Berlin (1998) · Zbl 0897.60056
[27] Prähofer, M.: Stochastic surface growth. Ph.D. thesis, Ludwig-Maximilians-Universität, München. http://edoc.ub.uni-muenchen.de/archive/00001381 (2003)
[28] Prähofer, M., Spohn, H.: Universal distributions for growth processes in 1+1 dimensions and random matrices. Phys. Rev. Lett. 84, 4882–4885 (2000)
[29] Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1071–1106 (2002) · Zbl 1025.82010
[30] Rákos, A., Schütz, G.: Current distribution and random matrix ensembles for an integrable asymmetric fragmentation process. J. Stat. Phys. 118, 511–530 (2005) · Zbl 1126.82330
[31] Rákos, A., Schütz, G.: Bethe Ansatz and current distribution for the TASEP with particle-dependent hopping rates. Markov Process. Relat. Fields 12, 323–334 (2006) · Zbl 1136.82350
[32] Rezakhanlou, F.: Hydrodynamic limit for attractive particle systems on \(\mathbb{Z}\) d . Commun. Math. Phys. 140, 417–448 (1991) · Zbl 0738.60098
[33] Rost, H.: Non-equilibrium behavior of a many particle system: density profile and local equilibrium. Z. Wahrsch. Verw. Gebiete 58, 41–53 (1981) · Zbl 0451.60097
[34] Sasamoto, T.: Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38, L549–L556 (2005)
[35] Schütz, G.M.: Exact solution of the master equation for the asymmetric exclusion process. J. Stat. Phys. 88, 427–445 (1997) · Zbl 0945.82508
[36] Schütz, G.M.: Exactly solvable models for many-body systems far from equilibrium. In: Domb, C., Lebowitz, J. (eds.) Phase Transitions and Critical Phenomena, vol. 19, pp. 1–251. Academic Press, London (2000)
[37] Soshnikov, A.: Determinantal random fields. In: Francoise, J.-P., Naber, G., Tsun, T.S. (eds.) Encyclopedia of Mathematical Physics, pp. 47–53. Elsevier, Oxford (2006)
[38] Spohn, H.: Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals. Physica A 369, 71–99 (2006)
[39] Stembridge, J.R.: Nonintersecting paths, Pfaffians, and plane partitions. Adv. Math. 83, 96–131 (1990) · Zbl 0790.05007
[40] Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994) · Zbl 0789.35152
[41] Tracy, C.A., Widom, H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727–754 (1996) · Zbl 0851.60101
[42] Viennot, G.: Une forme géométrique de la correspondence de Robinson-Schensted. In: Combinatoire et Représentation du Groupe Symétrique. Lecture Notes in Mathematics, vol. 579, pp. 29–58. Springer, Berlin (1977) · Zbl 0389.05016
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.