Multipoint distribution of periodic TASEP

Baik, Jinho; Liu, Zhipeng

doi:10.1090/jams/915

Multipoint distribution of periodic TASEP

Authors: Jinho Baik and Zhipeng Liu
Journal: J. Amer. Math. Soc. 32 (2019), 609-674
MSC (2010): Primary 60K35; Secondary 82C22
DOI: https://doi.org/10.1090/jams/915
Published electronically: January 8, 2019
Full-text PDF
View in AMS MathViewer New

Abstract | References | Similar Articles | Additional Information

Abstract: The height fluctuations of the models in the KPZ class are expected to converge to a universal process. The spatial process at equal time is known to converge to the Airy process or its variations. However, the temporal process, or more generally the two-dimensional space-time fluctuation field, is less well understood. We consider this question for the periodic TASEP (totally asymmetric simple exclusion process). For a particular initial condition, we evaluate the multitime and multilocation distribution explicitly in terms of a multiple integral involving a Fredholm determinant. We then evaluate the large-time limit in the so-called relaxation time scale.

[1] Gideon Amir, Ivan Corwin, and Jeremy Quastel, Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions, Comm. Pure Appl. Math. 64 (2011), no. 4, 466–537. MR 2796514, https://doi.org/10.1002/cpa.20347
[2] Jinho Baik, Percy Deift, and Kurt Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999), no. 4, 1119–1178. MR 1682248, https://doi.org/10.1090/S0894-0347-99-00307-0
[3] Jinho Baik, Patrik L. Ferrari, and Sandrine Péché, Limit process of stationary TASEP near the characteristic line, Comm. Pure Appl. Math. 63 (2010), no. 8, 1017–1070. MR 2642384, https://doi.org/10.1002/cpa.20316
[4] Jinho Baik and Zhipeng Liu, Fluctuations of TASEP on a ring in relaxation time scale, Comm. Pure Appl. Math. 71 (2018), no. 4, 747–813. MR 3772401, https://doi.org/10.1002/cpa.21702
[5] Jinho Baik and Zhipeng Liu, TASEP on a ring in sub-relaxation time scale, J. Stat. Phys. 165 (2016), no. 6, 1051–1085. MR 3575637, https://doi.org/10.1007/s10955-016-1665-y
[6] Alexei Borodin, Ivan Corwin, and Patrik Ferrari, Free energy fluctuations for directed polymers in random media in 1+1 dimension, Comm. Pure Appl. Math. 67 (2014), no. 7, 1129–1214. MR 3207195, https://doi.org/10.1002/cpa.21520
[7] Alexei Borodin and Patrik L. Ferrari, Large time asymptotics of growth models on space-like paths. I. PushASEP, Electron. J. Probab. 13 (2008), no. 50, 1380–1418. MR 2438811, https://doi.org/10.1214/EJP.v13-541
[8] Alexei Borodin, Patrik L. Ferrari, and Michael Prähofer, Fluctuations in the discrete TASEP with periodic initial configurations and the 𝐴𝑖𝑟𝑦₁ process, Int. Math. Res. Pap. IMRP 1 (2007), Art. ID rpm002, 47. MR 2334008
[9] Alexei Borodin, Patrik L. Ferrari, Michael Prähofer, and Tomohiro Sasamoto, Fluctuation properties of the TASEP with periodic initial configuration, J. Stat. Phys. 129 (2007), no. 5-6, 1055–1080. MR 2363389, https://doi.org/10.1007/s10955-007-9383-0
[10] Alexei Borodin, Patrik L. Ferrari, and Tomohiro Sasamoto, Transition between 𝐴𝑖𝑟𝑦₁ and 𝐴𝑖𝑟𝑦₂ processes and TASEP fluctuations, Comm. Pure Appl. Math. 61 (2008), no. 11, 1603–1629. MR 2444377, https://doi.org/10.1002/cpa.20234
[11] Ivan Corwin, The Kardar-Parisi-Zhang equation and universality class, Random Matrices Theory Appl. 1 (2012), no. 1, 1130001, 76. MR 2930377, https://doi.org/10.1142/S2010326311300014
[12] Ivan Corwin, Patrik L. Ferrari, and Sandrine Péché, Limit processes for TASEP with shocks and rarefaction fans, J. Stat. Phys. 140 (2010), no. 2, 232–267. MR 2659279, https://doi.org/10.1007/s10955-010-9995-7
[13] Ivan Corwin, Zhipeng Liu, and Dong Wang, Fluctuations of TASEP and LPP with general initial data, Ann. Appl. Probab. 26 (2016), no. 4, 2030–2082. MR 3543889, https://doi.org/10.1214/15-AAP1139
[14] Jacopo de Nardis and Pierre Le Doussal, Tail of the two-time height distribution for KPZ growth in one dimension, J. Stat. Mech. Theory Exp. 5 (2017), 053212, 72. MR 3664392, https://doi.org/10.1088/1742-5468/aa6bce
[15] Bernard Derrida and Joel L. Lebowitz, Exact large deviation function in the asymmetric exclusion process, Phys. Rev. Lett. 80 (1998), no. 2, 209–213. MR 1604439, https://doi.org/10.1103/PhysRevLett.80.209
[16] Victor Dotsenko, Two-time free energy distribution function in (1+1) directed polymers, J. Stat. Mech. Theory Exp. 6 (2013), P06017, 23. MR 3090739, https://doi.org/10.1088/1742-5468/2013/06/p06017
[17] Victor Dotsenko, Two-time distribution function in one-dimensional random directed polymers, J. Phys. A 48 (2015), no. 49, 495001, 18. MR 3434822, https://doi.org/10.1088/1751-8113/48/49/495001
[18] Victor Dotsenko, On two-time distribution functions in (1+1) random directed polymers, J. Phys. A 49 (2016), no. 27, 27LT01, 8. MR 3512099, https://doi.org/10.1088/1751-8113/49/27/27LT01
[19] Patrik L. Ferrari and Peter Nejjar, Anomalous shock fluctuations in TASEP and last passage percolation models, Probab. Theory Related Fields 161 (2015), no. 1-2, 61–109. MR 3304747, https://doi.org/10.1007/s00440-013-0544-6
[20] Patrik L. Ferrari and Herbert Spohn, On time correlations for KPZ growth in one dimension, SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 074, 23. MR 3529743, https://doi.org/10.3842/SIGMA.2016.074
[21] O. Golinelli and K. Mallick, Bethe ansatz calculation of the spectral gap of the asymmetric exclusion process, J. Phys. A 37 (2004), no. 10, 3321–3331. MR 2039850, https://doi.org/10.1088/0305-4470/37/10/001
[22] O. Golinelli and K. Mallick, Spectral gap of the totally asymmetric exclusion process at arbitrary filling, J. Phys. A 38 (2005), no. 7, 1419–1425. MR 2117259, https://doi.org/10.1088/0305-4470/38/7/001
[23] Shamik Gupta, Satya N. Majumdar, Claude Godrèche, and Mustansir Barma, Tagged particle correlations in the asymmetric simple exclusion process: finite-size effects, Phys. Rev. E (3) 76 (2007), no. 2, 021112, 17. MR 2365530, https://doi.org/10.1103/PhysRevE.76.021112
[24] L.-H. Gwa and H. Spohn,
Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation,
Phys. Rev. A 46 (1992), 844-854.
[25] T. Imamura and T. Sasamoto, Fluctuations of the one-dimensional polynuclear growth model with external sources, Nuclear Phys. B 699 (2004), no. 3, 503–544. MR 2098552, https://doi.org/10.1016/j.nuclphysb.2004.07.030
[26] Kurt Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), no. 2, 437–476. MR 1737991, https://doi.org/10.1007/s002200050027
[27] Kurt Johansson, Discrete polynuclear growth and determinantal processes, Comm. Math. Phys. 242 (2003), no. 1-2, 277–329. MR 2018275, https://doi.org/10.1007/s00220-003-0945-y
[28] Kurt Johansson, Two time distribution in Brownian directed percolation, Comm. Math. Phys. 351 (2017), no. 2, 441–492. MR 3613511, https://doi.org/10.1007/s00220-016-2660-5
[29] K. Johansson,
The two-time distribution in geometric last-passage percolation, 2018,
arXiv:1802.00729.
[30] Zhipeng Liu, Height fluctuations of stationary TASEP on a ring in relaxation time scale, Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018), no. 2, 1031–1057 (English, with English and French summaries). MR 3795076, https://doi.org/10.1214/17-AIHP831
[31] K. Matetski, J. Quastel, and D. Remenik,
The KPZ fixed point,
arXiv:1701.00018.
[32] V. S. Poghosyan and V. B. Priezzhev, Determinant solution for the TASEP with particle-dependent hopping probabilities on a ring, Markov Process. Related Fields 14 (2008), no. 2, 233–254. MR 2437530
[33] A. M. Povolotsky and V. B. Priezzhev, Determinant solution for the totally asymmetric exclusion process with parallel update. II. Ring geometry, J. Stat. Mech. Theory Exp. 8 (2007), P08018, 27. MR 2338272, https://doi.org/10.1088/1742-5468/2007/08/p08018
[34] Michael Prähofer and Herbert Spohn, Scale invariance of the PNG droplet and the Airy process, J. Statist. Phys. 108 (2002), no. 5-6, 1071–1106. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. MR 1933446, https://doi.org/10.1023/A:1019791415147
[35] V. Priezzhev,
Exact nonstationary probabilities in the asymmetric exclusion process on a ring,
Phys. Rev. Lett. 91 (2003), no. 5, 050601.
[36] S. Prolhac,
Finite-time fluctuations for the totally asymmetric exclusion process,
Phys. Rev. Lett. 116 (2016), 090601.
[37] J. Quastel and D. Remenik,
How flat is flat in random interface growth?
arXiv:1606.09228.
[38] A. Rákos and G. M. Schütz, Current distribution and random matrix ensembles for an integrable asymmetric fragmentation process, J. Stat. Phys. 118 (2005), no. 3-4, 511–530. MR 2123646, https://doi.org/10.1007/s10955-004-8819-z
[39] T. Sasamoto, Spatial correlations of the 1D KPZ surface on a flat substrate, J. Phys. A 38 (2005), no. 33, L549–L556. MR 2165697, https://doi.org/10.1088/0305-4470/38/33/L01
[40] Gunter M. Schütz, Exact solution of the master equation for the asymmetric exclusion process, J. Statist. Phys. 88 (1997), no. 1-2, 427–445. MR 1468391, https://doi.org/10.1007/BF02508478
[41] Craig A. Tracy and Harold Widom, Asymptotics in ASEP with step initial condition, Comm. Math. Phys. 290 (2009), no. 1, 129–154. MR 2520510, https://doi.org/10.1007/s00220-009-0761-0