Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 052, 11 pages      arXiv:0904.3680
Contribution to the Proceedings of the XVIIth International Colloquium on Integrable Systems and Quantum Symmetries

Determinantal Representation of the Time-Dependent Stationary Correlation Function for the Totally Asymmetric Simple Exclusion Model

Nikolay M. Bogoliubov
St. Petersburg Department of V.A. Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia

Received October 30, 2008, in final form April 14, 2009; Published online April 23, 2009

The basic model of the non-equilibrium low dimensional physics the so-called totally asymmetric exclusion process is related to the 'crystalline limit' (q → ∞) of the SUq(2) quantum algebra. Using the quantum inverse scattering method we obtain the exact expression for the time-dependent stationary correlation function of the totally asymmetric simple exclusion process on a one dimensional lattice with the periodic boundary conditions.

Key words: quantum inverse method; algebraic Bethe ansatz; asymmetric exclusion process.

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  1. Faddeev L.D., Quantum completely integrable models in field theory, Soviet Sci. Rev. Sect. C: Math. Phys. Rev., Vol. 1, Harwood Academic, Chur, 1980, 107-155.
  2. Kulish P.P., Sklyanin E.K., Quantum spectral transform method. Recent developments, Lecture Notes in Phys., Vol. 151, Springer, Berlin - New York, 1982, 61-119.
  3. Korepin V.E., Bogoliubov N.M., Izergin A.G., Quantum inverse scattering method and correlation functions, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1993.
  4. Drinfel'd V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 798-820.
  5. Faddeev L.D., Reshetikhin N.Yu., Takhtajan L.A., Quantisation of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990), 193-225.
  6. Kashivara M., Crystalizing the q-analogue of universal enveloping algebras, Comm. Math. Phys. 133 (1990), 249-260.
  7. Stanley R., Enumerative combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, Vol. 62, Cambridge University Press, Cambridge, 1999.
  8. Bressoud D.M., Proofs and confirmations. The story of the alternating sign matrix conjecture, Cambridge University Press, Cambridge, 1999.
  9. Evans M.R., Blythe R.A., Nonequilibrium dynamics in low-dimensional systems, Phys. A 313 (2002), 110-152, cond-mat/0110630.
  10. Schütz G., Exactly solvable models for many-body systems far from equilibrium, Phase Transitions and Critical Phenomena, Vol. 19, Editors C. Domb and J.L. Lebowitz, Academic Press, San Diego, CA, 2001, 1-251.
  11. Korepin V., Dual field formulation of quantum integrable models, Comm. Math. Phys. 113 (1987), 177-190.
  12. Bogoliubov N.M., Izergin A.G., Kitanine N.A., Correlators of the phase model, Phys. Lett. A 231 (1997), 347-352, solv-int/9612002.
  13. Bogoliubov N.M., Izergin A.G., Kitanine N.A., Correlation functions for a strongly correlated boson system, Nuclear Phys. B 516 (1998), 501-528, solv-int/9710002.
  14. Gwa L.-H., Spohn H., Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation, Phys. Rev. A 46 (1992), 844-854.
  15. Derrida B., Lebowitz J., Exact large deviation function in the asymmetric exclusion process, Phys. Rev. Lett. 80 (1998), 209-213, cond-mat/9809044.
  16. Golinelli O., Mallick K., Bethe ansatz calculation of the spectral gap of the asymmetric exclusion process, J. Phys. A: Math. Gen. 37 (2004), 3321-3331, cond-mat/0312371.
  17. Golinelli O., Mallick K., Spectral gap of the totally asymmetric exclusion process at arbitrary filling, J. Phys. A: Math. Gen. 38 (2005), 1419-1425, cond-mat/0411505.
  18. de Gier J., Essler F.H.L., Bethe ansatz solution of the asymmetric exclusion process with open boundaries, Phys. Rev. Lett. 95 (2005), 240601, 4 pages, cond-mat/0508707.
  19. Prolhac S., Mallick K., Current fluctuations in the exclusion process and Bethe ansatz, J. Phys. A: Math. Theor. 41 (2008), 175002, 20 pages, arXiv:0801.4659.
  20. Noh J.D., Kim D., Interacting domain walls and the five-vertex model, Phys. Rev. E 49 (1995), 1943-1961, cond-mat/9312001.
  21. Kim D., Bethe Ansatz solution for crossover scaling functions of the asymmetric XXZ chain and the Kardar-Parisi-Zhang-type growth model, Phys. Rev. E 52 (1995), 3512-3524, cond-mat/9503169.
  22. Lee D.S., Kim D., Large deviation function of the partially asymmetric exclusion process, Phys. Rev. E 59 (1999), 6476-6482, cond-mat/9902001.
  23. Bogoliubov N.M., Nassar T., On the spectrum of the non-Hermitian phase-difference model, Phys. Lett. A 234 (1997), 345-350.
  24. Golinelli O., Mallick K., The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics, J. Phys. A: Math. Gen. 39 (2006), 12679-12705, cond-mat/0611701.
  25. Schütz G., Exact solution of the master equation for the asymmetric exclusion process, J. Statist. Phys. 88 (1997), 427-445.
  26. Borodin A., Ferrari P., Prahofer M., Sasamoto T., Fluctuation properties of the TASEP with periodic initial configuration, J. Statist. Phys. 129 (2007), 1055-1080, math-ph/0608056.
  27. Sasamoto T., Spatial correlations of the 1D KPZ surface on a flat substrate, J. Phys. A: Math. Gen. 38 (2005), L549-L556, cond-mat/0504417.
  28. Prähofer M., Spohn H., Current fluctuations for the totally asymmetric simple exclusion process, in In and Out of Equilibrium (Mambucaba, 2000), Editor V. Sidoravicius, Progr. Probab., Vol. 51, Birkhäuser Boston, Boston, MA, 2002, 185-204, cond-mat/0101200.
  29. Prähofer M., Spohn H., Exact scaling functions for one-dimensional stationary KPZ growth, J. Statist. Phys. 115 (2004), 255-279.
  30. Schütz G., Duality relations for asymmetric exclusion processes, J. Statist. Phys. 86 (1997), 1265-1287.
  31. Bogoliubov N.M., Five vertex model with the fixed boundary conditions, Algebra i Analiz 21 (2009), 58-78.
  32. Macdonald I.G., Symmetric functions and Hall polynomials, Oxford Mathematical Monographs. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
  33. Bogoliubov N.M., Boxed plane partitions as an exactly solvable boson model, J. Phys. A: Math. Gen. 38 (2005), 9415-9430, cond-mat/0503748.
  34. Bogoliubov N.M., Enumeration of plane partitions and the algebraic Bethe ansatz, Theoret. and Math. Phys. 150 (2007), 165-174.
  35. Bogoliubov N.M., Four-vertex model and random tilings, Theoret. and Math. Phys. 155 (2008), 523-535, arXiv:0711.0030.
  36. Tsilevich N., Quantum inverse method for the q-boson model, and symmetric functions, Funct. Anal. Appl. 40 (2006), 207-217, math-ph/0510073.
  37. Shigechi K., Uchiyama M., Boxed skew plane partition and integrable phase model, J. Phys. A: Math. Gen. 38 (2005), 10287-10306, cond-mat/0508090.
  38. Bogoliubov N.M., Integrable models for vicious and friendly walkers, J. Math. Sci. (N.Y.) 143 (2007), 2729-2737.
  39. Ferrari P.L., Spohn H., Step functions for a faceted crystal, J. Statist. Phys. 113 (2003), 1-46.
  40. Rajesh R., Dhar D., An exactly solvable anisotropic directed percolation model in three dimensions, Phys. Rev. Lett. 81 (1998), 1646-1649, cond-mat/9808023.
  41. Kardar M., Parisi G., Zhang Y.Z., Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889-892.

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