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Current fluctuations for TASEP: A proof of the Prähofer-Spohn conjecture. (English) Zbl 1208.82036

Summary: We consider the family of two-sided Bernoulli initial conditions for TASEP which, as the left and right densities \((\rho _{ - }, \rho _{+})\) are varied, give rise to shock waves and rarefaction fans-the two phenomena which are typical to TASEP. We provide a proof of Conjecture 7.1 in [M. Prähofer and H. Spohn, in: In and out of equilibrium. Probability with a physics flavor. Papers from the 4th Brazilian school of probability, Mambucaba, Brazil, August 14–19, 2000. Boston: Birkhäuser. Prog. Probab. 51, 185–204 (2002; Zbl 1015.60093)] which characterizes the order of and scaling functions for the fluctuations of the height function of two-sided TASEP in terms of the two densities \(\rho _{ - }, \rho _{+}\) and the speed \(y\) around which the height is observed.
In proving this theorem for TASEP, we also prove a fluctuation theorem for a class of corner growth processes with external sources, or equivalently for the last passage time in a directed last passage percolation model with two-sided boundary conditions: \(\rho _{ - }\) and \(1 - \rho _{+}\). We provide a complete characterization of the order of and the scaling functions for the fluctuations of this model’s last passage time \(L(N, M)\) as a function of three parameters: the two boundary/source rates \(\rho _{ - }\) and \(1 - \rho _{+}\), and the scaling ratio \(\gamma ^{2}=M/N\). The proof of this theorem draws on the results of [Comm. Math. Phys. 265 (2006) 1-44] and extensively on the work of [J. Baik, G. Ben Arous and S. Péché, Ann. Probab. 33, No. 5, 1643–1697 (2005; Zbl 1086.15022)] on finite rank perturbations of Wishart ensembles in random matrix theory.

MSC:

82C22 Interacting particle systems in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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[1] Baik, J. (2006). Painlevé formulas of the limiting distributions for nonnull complex sample covariance matrices. Duke Math. J. 133 205-235. · Zbl 1139.33006
[2] Baik, J., Ben Arous, G. and Péché, S. (2005). Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 1643-1697. · Zbl 1086.15022
[3] Baik, J. and Rains, E. M. (2000). Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 100 523-541. · Zbl 0976.82043
[4] Balázs, M., Cator, E. and Seppäläinen, T. (2006). Cube root fluctuations for the corner growth model associated to the exclusion process. Electron. J. Probab. 11 1094-1132. · Zbl 1139.60046
[5] Balázs, M. and Seppäläinen, T. (2009). Fluctuation bounds for the asymmetric simple exclusion process. ALEA Lat. Am. J. Probab. Math. Stat. 6 1-24. · Zbl 1160.60333
[6] Balázs, M. and Seppäläinen, T. (2010). Order of current variance and diffusivity in the asymmetric simple exclusion process. Ann. of Math. (2) 171 1237-1265. · Zbl 1200.60083
[7] Borodin, A., Ferrari, P. L. and Sasamoto, T. (2009). Two speed TASEP. J. Stat. Phys. 137 936-977. · Zbl 1183.82062
[8] Derrida, B. and Gerschenfeld, A. (2009). Current fluctuations of the one dimensional symmetric simple exclusion process with step initial condition. J. Stat. Phys. 136 1-15. · Zbl 1173.82019
[9] Ferrari, P. A. (1992). Shock fluctuations in asymmetric simple exclusion. Probab. Theory Related Fields 91 81-101. · Zbl 0744.60117
[10] Ferrari, P. A. and Fontes, L. R. G. (1994). Current fluctuations for the asymmetric simple exclusion process. Ann. Probab. 22 820-832. · Zbl 0806.60099
[11] Ferrari, P. A. and Fontes, L. R. G. (1994). Shock fluctuations in the asymmetric simple exclusion process. Probab. Theory Related Fields 99 305-319. · Zbl 0801.60094
[12] Ferrari, P. A. and Kipnis, C. (1995). Second class particles in the rarefaction fan. Ann. Inst. H. Poincaré Probab. Statist. 31 143-154. · Zbl 0813.60095
[13] Ferrari, P. A., Kipnis, C. and Saada, E. (1991). Microscopic structure of travelling waves in the asymmetric simple exclusion process. Ann. Probab. 19 226-244. · Zbl 0725.60113
[14] Ferrari, P. L. and Spohn, H. (2006). Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Comm. Math. Phys. 265 1-44. · Zbl 1118.82032
[15] Imamura, T. and Sasamoto, T. (2004). Fluctuations of the one-dimensional polynuclear growth model with external sources. Nuclear Phys. B 699 503-544. · Zbl 1123.82352
[16] Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437-476. · Zbl 0969.15008
[17] Liggett, T. M. (1999). Stochastic Interacting Systems : Contact , Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 324 . Springer, Berlin. · Zbl 0949.60006
[18] Liggett, T. M. (2005). Interacting Particle Systems . Springer, Berlin. · Zbl 1103.82016
[19] Mountford, T. and Guiol, H. (2005). The motion of a second class particle for the TASEP starting from a decreasing shock profile. Ann. Appl. Probab. 15 1227-1259. · Zbl 1069.60091
[20] Nagao, T. and Sasamoto, T. (2004). Asymmetric simple exclusion process and modified random matrix ensembles. Nuclear Phys. B 699 487-502. · Zbl 1123.82345
[21] Onatski, A. (2008). The Tracy-Widom limit for the largest eigenvalues of singular complex Wishart matrices. Ann. Appl. Probab. 18 470-490. · Zbl 1141.60009
[22] Prähofer, M. and Spohn, H. (2002). Current fluctuations for the totally asymmetric simple exclusion process. In In and Out of Equilibrium ( Mambucaba , 2000). Progress in Probability 51 185-204. Birkhäuser, Boston, MA. · Zbl 1015.60093
[23] Prähofer, M. and Spohn, H. (2002). Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108 1071-1106. · Zbl 1025.82010
[24] Quastel, J. and Valko, B. (2007). t 1\?3 superdiffusivity of finite-range asymmetric exclusion processes on \Bbb Z. Comm. Math. Phys. 273 379-394. · Zbl 1127.60091
[25] Rezakhanlou, F. (2002). A central limit theorem for the asymmetric simple exclusion process. Ann. Inst. H. Poincaré Probab. Statist. 38 437-464. · Zbl 1001.60031
[26] Rost, H. (1981). Nonequilibrium behaviour of a many particle process: Density profile and local equilibria. Z. Wahrsch. Verw. Gebiete 58 41-53. · Zbl 0451.60097
[27] Sasamoto, T. (2007). Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques. J. Stat. Mech. Theory Exp. P07007.
[28] Seppäläinen, T. (1998). Hydrodynamic scaling, convex duality and asymptotic shapes of growth models. Markov Process. Related Fields 4 1-26. · Zbl 0906.60082
[29] Seppäläinen, T. (2002). Diffusive fluctuations for one-dimensional totally asymmetric interacting random dynamics. Comm. Math. Phys. 229 141-182. · Zbl 1043.82028
[30] Tracy, C. A. and Widom, H. (2009). Asymptotics in ASEP with step initial condition. Comm. Math. Phys. 290 129-154. · Zbl 1184.60036
[31] Tracy, C. and Widom, H. (2009). Total current fluctuations in ASEP. J. Math. Phys. 50 095204. · Zbl 1241.82051
[32] Tracy, C. A. and Widom, H. (2008). A Fredholm determinant representation in ASEP. J. Stat. Phys. 132 291-300. · Zbl 1144.82045
[33] Tracy, C. A. and Widom, H. (2008). Integral formulas for the asymmetric simple exclusion process. Comm. Math. Phys. 279 815-844. · Zbl 1148.60080
[34] Tracy, C. A. and Widom, H. (2009). On ASEP with step Bernoulli initial condition. J. Stat. Phys. 137 825-838. · Zbl 1188.82043
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