×

Anomalous shock fluctuations in TASEP and last passage percolation models. (English) Zbl 1311.60116

The totally asymmetric simple exclusion process (TASEP) is an interacting particle system on the integers \(\mathbb Z\). Particles jump at constant rate onto the nearest-neighbor site at the right, provided that it is unoccupied by a particle. The particle density at time \(t\) solves an approximation of Burger’s partial differential equation, and the discontinuities of the solution are called shocks. Earlier works on TASEP models started from product measures, which are the only invariant measures. For initial product measures with a shock at the origin, fluctuations of the shock location are Gaussian of order \(t^{1/2}\) in the asymptotic limit of time. Exploiting the earlier established connection between last passage percolation (LLP) on the integer plane \(\mathbb{Z}^2\) and the TASEP, the authors derive a generic theorem, which dictates that “the distribution function of a generic LLP is the product of two distribution functions corresponding to two simpler last passage problems.” Their generic theorem is applied to a variety of LLP and TASEP models through the verification of the assumptions of the theorem. For a certain TASEP model that motivated their work, the model is started from a hybrid configuration \((\dots, 0, 1, 0, 1, \dots)\) interpreting 1 as occupied. The particles attempt to jump at two different rates depending on whether they are initially located at the left or the right of the origin. Shock fluctuations of order \(t^{1/3}\) are expressed in the long term as a product two Airy\({}_1\) processes, whose one-dimensional marginal distribution is described by the Tracy-Widom distribution of the largest eigenvalue in the Gaussian orthogonal ensemble of random matrices. This accumulates evidence for the universality of the Airy\({}_1\) process, for which there is a certain lack of findings. In other corollaries of the generic theorem, the Airy\({}_2\) process is involved, whose one-dimensional marginal distribution is described by the Tracy-Widom distribution of the largest eigenvalue in the Gaussian unitary ensemble.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Andjel, E.D., Vares, M.E.: Hydrodynamic equations for attractive particle systems on \[\mathbb{Z}\] Z. J. Stat. Phys. 47, 265-288 (1987) · Zbl 0685.58043 · doi:10.1007/BF01009046
[2] 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 (2006) · Zbl 1086.15022 · doi:10.1214/009117905000000233
[3] Baik, J., Ferrari, P.L., Péché, S.: Limit process of stationary TASEP near the characteristic line. Comm. Pure Appl. Math. 63, 1017-1070 (2010) · Zbl 1194.82067
[4] Baik, J., Ferrari, P.L., Péché, S.: Convergence of the two-point function of the stationary TASEP, arXiv:1209.0116 (2012) · Zbl 1355.82024
[5] 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
[6] Baik, J., Rains, E.M.: Symmetrized Random Permutations, Random Matrix Models and Their Applications, pp. 1-19. Cambridge University Press, Cambridge (2001) · Zbl 0989.60010
[7] van Beijeren, H.: Fluctuations in the motions of mass and of patterns in one-dimensional driven diffusive systems. J. Stat. Phys. 63, 47-58 (1991) · doi:10.1007/BF01026591
[8] Belitsky, V., Schütz, G.M.: Microscopic structure of shocks and antishocks in the ASEP conditioned on low current. J. Stat. Phys. 152, 93-111 (2013) · Zbl 1281.82025 · doi:10.1007/s10955-013-0758-0
[9] Ben Arous, G., Corwin, I.: Current fluctuations for TASEP: a proof of the Prähofer-Spohn conjecture. Ann. Probab. 39, 104-138 (2011) · Zbl 1208.82036 · doi:10.1214/10-AOP550
[10] Borodin, A., Ferrari, P.L.: Large time asymptotics of growth models on space-like paths I: PushASEP. Electron. J. Probab. 13, 1380-1418 (2008) · Zbl 1187.82084 · doi:10.1214/EJP.v13-541
[11] Borodin, A., Ferrari, P.L., Prähofer, M.: Fluctuations in the discrete TASEP with periodic initial configurations and the Airy \[_11\] process. Int. Math. Res. Papers 2007, rpm002 (2007) · Zbl 1136.82321
[12] Borodin, A., Ferrari, P.L., Prähofer, M., Sasamoto, T.: Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129, 1055-1080 (2007) · Zbl 1136.82028 · doi:10.1007/s10955-007-9383-0
[13] Borodin, A., Ferrari, P.L., Sasamoto, T.: Transition between Airy \[_11\] and Airy \[_22\] processes and TASEP fluctuations. Comm. Pure Appl. Math. 61, 1603-1629 (2008) · Zbl 1214.82062 · doi:10.1002/cpa.20234
[14] Borodin, A., Ferrari, P.L., Sasamoto, T.: Two speed TASEP. J. Stat. Phys. 137, 936-977 (2009) · Zbl 1183.82062 · doi:10.1007/s10955-009-9837-7
[15] Corwin, I.: The Kardar-Parisi-Zhang equation and universality class. Random Matrices Theory Appl. 1 (2012). doi:10.1142/S2010326311300014 · Zbl 1247.82040
[16] Corwin, I., Ferrari, P.L., Péché, S.: Universality of slow decorrelation in KPZ models. Ann. Inst. H. Poincaré Probab. Statist. 48, 134-150 (2012) · Zbl 1247.82041 · doi:10.1214/11-AIHP440
[17] Derrida, B., Janowsky, S.A., Lebowitz, J.L., Speer, E.R.: Exact solution of the totally asymmetric simple exclusion process: shock profiles. J. Stat. Phys. 73, 813-842 (1993) · Zbl 1102.60320 · doi:10.1007/BF01052811
[18] Ferrari, P.A.: The simple exclusion process as seen from a tagged particle. Ann. Probab. 14, 1277-1290 (1986) · Zbl 0628.60103 · doi:10.1214/aop/1176992369
[19] Ferrari, P.A.: Shock fluctuations in asymmetric simple exclusion. Probab. Theory Relat. Fields 91, 81-101 (1992) · Zbl 0744.60117 · doi:10.1007/BF01194491
[20] Ferrari, P.A., Fontes, L.: Shock fluctuations in the asymmetric simple exclusion process. Probab. Theory Relat. Fields 99, 305-319 (1994) · Zbl 0801.60094 · doi:10.1007/BF01199027
[21] Ferrari, P.A., Kipnis, C., Saada, E.: Microscopic structure of travelling waves in the asymmetric simple exclusion process. Ann. Probab. 19, 226-244 (1991) · Zbl 0725.60113 · doi:10.1214/aop/1176990542
[22] Ferrari, P.L.: Slow decorrelations in KPZ growth. J. Stat. Mech. P07022 (2008) · Zbl 1459.82175
[23] Ferrari, P.L.: The universal Airy \[_11\] and Airy \[_22\] processes in the Totally Asymmetric Simple Exclusion Process. In: Baik, J., Kriecherbauer, T., Li, L.-C., McLaughlin K., Tomei, C. (eds.) Integrable Systems and Random Matrices: In Honor of Percy Deifts, Contemporary Mathematics, vol. 458, pp. 321-332. American Mathematical Society (2008). · Zbl 1145.82332
[24] Ferrari, P.L., Spohn, H.: A determinantal formula for the GOE Tracy-Widom distribution. J. Phys. A 38, L557-L561 (2005) · doi:10.1088/0305-4470/38/33/L02
[25] Ferrari, P.L., Spohn, H.: Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Comm. Math. Phys. 265, 1-44 (2006) · Zbl 1118.82032 · doi:10.1007/s00220-006-1549-0
[26] Gärtner, J., Presutti, E.: Shock fluctuations in a particle system. Ann. Inst. H. Poincaré (A) 53, 1-14 (1990) · Zbl 0705.76054
[27] Johansson, K.: Shape fluctuations and random matrices. Comm. Math. Phys. 209, 437-476 (2000) · Zbl 0969.15008 · doi:10.1007/s002200050027
[28] Johansson, K.: Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Related Fields 116, 445-456 (2000) · Zbl 0960.60097 · doi:10.1007/s004400050258
[29] Johansson, K.: Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242, 277-329 (2003) · Zbl 1031.60084
[30] Kardar, M., 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
[31] Liggett, T.M.: Coupling the simple exclusion process. Ann. Probab. 4, 339-356 (1976) · Zbl 0339.60091 · doi:10.1214/aop/1176996084
[32] Liggett, T.M.: Stochastic interacting systems: contact, voter and exclusion processes. Springer, Berlin (1999) · Zbl 0949.60006 · doi:10.1007/978-3-662-03990-8
[33] Prähofer, M., Spohn, H.: Current fluctuations for the totally asymmetric simple exclusion process. In: Sidoravicius, V. (ed.) In and Out of Equilibrium, Progress in Probability, vol. 51, pp. 185-204. Birkhäuser, Basel (2002) · Zbl 1015.60093
[34] Sasamoto, T.: 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
[35] Spohn, H.: Large Scale Dynamics of Interacting Particles, Texts and Monographs in Physics. Springer, Heidelberg (1991) · doi:10.1007/978-3-642-84371-6
[36] Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159, 151-174 (1994) · Zbl 0789.35152 · doi:10.1007/BF02100489
[37] Tracy, C.A., Widom, H.: On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177, 727-754 (1996) · Zbl 0851.60101 · doi:10.1007/BF02099545
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.