Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Limit law of a second class particle in TASEP with non-random initial condition

P. L. Ferrari, P. Ghosal, and P. Nejjar

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Abstract

We consider the totally asymmetric simple exclusion process (TASEP) with non-random initial condition and density $\lambda$ on $\mathbb{Z}_{-}$ and $\rho$ on $\mathbb{Z}_{+}$, and a second class particle initially at the origin. For $\lambda<\rho$, there is a shock and the second class particle moves with speed $1-\lambda-\rho$. For large time $t$, we show that the position of the second class particle fluctuates on the $t^{1/3}$ scale and determine its limiting law. We also obtain the limiting distribution of the number of steps made by the second class particle until time $t$.

Résumé

On considère le processus d’exclusion simple totalement asymétrique avec des condition initiales déterministes, densité $\lambda$ sur $\mathbb{Z}_{-}$ et $\rho$ sur $\mathbb{Z}_{+}$. Initialement on place une particule de deuxième classe à l’origine. Si $\lambda<\rho$, un choc est créé et la particule de deuxième classe le suit avec vitesse $1-\lambda-\rho$. Dans la limite $t\to\infty$, on démontre que les fluctuations de la position de la particule de deuxième classe sont de l’ordre $t^{1/3}$ et on obtient sa loi limite. On détermine aussi la loi limite du nombre de sauts faits par la particule de deuxième classe jusqu’à l’instant $t$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1203-1225.

Dates
Received: 26 October 2017
Revised: 4 May 2018
Accepted: 21 May 2018
First available in Project Euclid: 25 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1569398867

Digital Object Identifier
doi:10.1214/18-AIHP916

Mathematical Reviews number (MathSciNet)
MR4010933

Subjects
Primary: 82C22: Interacting particle systems [See also 60K35] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Keywords
Exclusion process Second class particle Shock Scaling limit KPZ universality class

Citation

Ferrari, P. L.; Ghosal, P.; Nejjar, P. Limit law of a second class particle in TASEP with non-random initial condition. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1203--1225. doi:10.1214/18-AIHP916. https://projecteuclid.org/euclid.aihp/1569398867


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References

  • [1] J. Baik, G. Ben Arous and S. Péché. Phase transition of the largest eigenvalue for non-null complex sample covariance matrices. Ann. Probab. 33 (2005) 1643–1697.
  • [2] J. Baik, P. L. Ferrari and S. Péché. Convergence of the two-point function of the stationary TASEP. In Singular Phenomena and Scaling in Mathematical Models 91–110. Springer, Berlin, 2014.
  • [3] M. Balázs, E. Cator and T. Seppäläinen. Cube root fluctuations for the corner growth model associated to the exclusion process. Electron. J. Probab. 11 (2006) 1094–1132.
  • [4] A. Borodin and V. Gorin. Lectures on integrable probability, 2012. Available at arXiv:1212.3351.
  • [5] A. Borodin and S. Shlosman. Gibbs ensembles of nonintersecting paths. Comm. Math. Phys. 293 (2010) 145–170.
  • [6] E. Cator and L. Pimentel. On the local fluctuations of last-passage percolation models. Stochastic Process. Appl. 125 (2015) 879–903.
  • [7] I. Corwin. The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 1 (2012) 1130001.
  • [8] I. Corwin, P. L. Ferrari and S. Péché. Universality of slow decorrelation in KPZ models. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 134–150.
  • [9] P. A. Ferrari. Shock fluctuations in asymmetric simple exclusion. Probab. Theory Related Fields 91 (1992) 81–101.
  • [10] P. A. Ferrari and L. Fontes. Shock fluctuations in the asymmetric simple exclusion process. Probab. Theory Related Fields 99 (1994) 305–319.
  • [11] P. A. Ferrari and C. Kipnis. Second class particles in the rarefaction fan. Ann. Inst. Henri Poincaré 31 (1995) 143–154.
  • [12] P. A. Ferrari, J. B. Martin and L. P. R. Pimentel. A phase transition for competition interfaces. Ann. Appl. Probab. 19 (2009) 281–317.
  • [13] P. A. Ferrari and L. P. R. Pimentel. Competition interfaces and second class particles. Ann. Probab. 33 (2005) 1235–1254.
  • [14] P. L. Ferrari. Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues. Comm. Math. Phys. 252 (2004) 77–109.
  • [15] P. L. Ferrari. Slow decorrelations in KPZ growth. J. Stat. Mech. (2008) P07022.
  • [16] P. L. Ferrari. From interacting particle systems to random matrices. J. Stat. Mech. (2010) P10016.
  • [17] P. L. Ferrari and P. Nejjar. Anomalous shock fluctuations in TASEP and last passage percolation models. Probab. Theory Related Fields 161 (2015) 61–109.
  • [18] P. L. Ferrari and P. Nejjar. Fluctuations of the competition interface in presence of shocks. ALEA Lat. Am. J. Probab. Math. Stat. 14 (2017) 299–325.
  • [19] P. L. Ferrari and A. Occelli. Universality of the GOE Tracy–Widom distribution for TASEP with arbitrary particle density. Preprint, 2017. Available at arXiv:1704.01291.
  • [20] P. L. Ferrari and H. Spohn. Scaling limit for the space–time covariance of the stationary totally asymmetric simple exclusion process. Comm. Math. Phys. 265 (2006) 1–44.
  • [21] P. L. Ferrari and H. Spohn. Random growth models. In The Oxford Handbook of Random Matrix Theory 782–801. G. Akemann, J. Baik and P. Di Francesco (Eds). Oxford Univ. Press, Oxford, 2011.
  • [22] J. Gärtner and E. Presutti. Shock fluctuations in a particle system. Ann. Inst. Henri Poincaré A, Phys. Théor. 53 (1990) 1–14.
  • [23] K. Johansson. Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000) 437–476.
  • [24] M. Kardar, G. Parisi and Y. Z. Zhang. Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 (1986) 889–892.
  • [25] T. M. Liggett. Interacting Particle Systems. Springer Verlag, Berlin, 1985.
  • [26] T. M. Liggett. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer Verlag, Berlin, 1999.
  • [27] T. Mountford and H. Guiol. The motion of a second class particle for the TASEP starting from a decreasing shock profile. Ann. Appl. Probab. 15 (2005) 1227–1259.
  • [28] P. Nejjar. Transition to shocks in TASEP and decoupling of last passage times. Preprint, 2017. Available at arXiv:1705.08836.
  • [29] L. P. R. Pimentel. Local behavior of airy processes. Preprint, 2017. Available at arXiv:1704.01903.
  • [30] M. Prähofer and H. Spohn. Universal distributions for growth processes in $1+1$ dimensions and random matrices. Phys. Rev. Lett. 84 (2000) 4882–4885.
  • [31] M. Prähofer and H. Spohn. Current fluctuations for the totally asymmetric simple exclusion process. In In and Out of Equilibrium, V. Sidoravicius (Ed.). Progress in Probability. Birkhäuser, Basel, 2002.
  • [32] J. Quastel. Introduction to KPZ. In Current Developments in Mathematics 125–194, 2012.
  • [33] J. Quastel and H. Spohn. The one-dimensional KPZ equation and its universality class. J. Stat. Phys. 160 (2015) 965–984.
  • [34] C. A. Tracy and H. Widom. On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 (1996) 727–754.