Electronic Journal of Probability

Large time asymptotics of growth models on space-like paths I: PushASEP

Alexei Borodin and Patrik Ferrari

Full-text: Open access


We consider a new interacting particle system on the one-dimensional lattice that interpolates between TASEP and Toom's model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way. We prove that for flat and step initial conditions, the large time fluctuations of theheight function of the associated growth model along any space-like path are described by the Airy<sub>1</sub> and Airy<sub>2</sub> processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle's trajectory as special cases.

Article information

Electron. J. Probab. Volume 13 (2008), paper no. 50, 1380-1418.

Accepted: 25 August 2008
First available in Project Euclid: 1 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 82C22: Interacting particle systems [See also 60K35]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 15A52

stochastic growth KPZ determinantal processes Airy processes

This work is licensed under a Creative Commons Attribution 3.0 License.


Borodin, Alexei; Ferrari, Patrik. Large time asymptotics of growth models on space-like paths I: PushASEP. Electron. J. Probab. 13 (2008), paper no. 50, 1380--1418. doi:10.1214/EJP.v13-541. https://projecteuclid.org/euclid.ejp/1464819123.

Export citation


  • Pocketbook of mathematical functions. Abridged edition of Handbook of mathematical functions edited by Milton Abramowitz and Irene A. Stegun. Material selected by Michael Danos and Johann Rafelski. Verlag Harri Deutsch, Thun, 1984. 468 pp. ISBN: 3-87144-818-4 (85j:00005b)
  • M. Alimohammadi, V. Karimipour, M. Khorrami. A two-parametric family of asymmetric exclusion processes and its exact solution. J. Statist. Phys. 97 (1999), no. 1-2, 373–394.
  • Alexei Borodin, Patrik L. Ferrari, Michael Prähofer. Fluctuations in the discrete TASEP with periodic initial configurations and the Airy1 process. Int. Math. Res. Pap. IP 2007, no. 1, Art. ID rpm002, 47 pp.
  • Alexei Borodin, Patrik L. Ferrari, Michael Prähofer, Tomohiro Sasamoto. Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129 (2007), no. 5-6, 1055–1080.
  • Alexei Borodin, Patrik L. Ferrari, Tomohiro Sasamoto. Transition between Airy1 and Airy2 processes and TASEP fluctuations. Comm. Pure Appl. Math. online first (2007).
  • Alexei Borodin, Patrik L. Ferrari, Tomohiro Sasamoto. Large time asymptotics of growth models on space-like paths II: PNG and parallel TASEP. Comm. Math. Phys. 283 (2008), 417–449.
  • Alexei Borodin, Grigori Olshanski. Stochastic dynamics related to Plancherel measure on partitions. Representation theory, dynamical systems, and asymptotic combinatorics, 9–21, Amer. Math. Soc. Transl. Ser. 2, 217, Amer. Math. Soc., Providence, RI, 2006.
  • Alexei Borodin, Eric M. Rains. Eynard-Mehta theorem, Schur process, and their Pfaffian analogs. J. Stat. Phys. 121 (2005), no. 3-4, 291–317.
  • B. Derrida, J.L. Lebowitz, E.R. Speer, H. Spohn. Dynamics of an anchored Toom interface. J. Phys. A 24 (1991), no. 20, 4805–4834.
  • A.B. Dieker, J. Warren. Determinantal transition kernels for some interacting particles on the line. arXiv:0707.1843; To appear in Ann. Inst. H. Poincaré (B) (2007).
  • Patrik L. Ferrari. The universal Airy1 and Airy2 processes in the Totally Asymmetric Simple Exclusion Process. Integrable Systems and Random Matrices: In Honor of Percy Deift edited by Jinho Baik, Thomas Kriecherbauer, Luen-Chau Li, Kenneth D. T-R McLaughlin, and Carlos Tomei. Contemporary Math., Amer. Math. Soc., 2008, pp. 321–332.
  • J. Ben Hough, Manjunath Krishnapur, Yuval Peres, Bálint Virág. Determinantal processes and independence. Probab. Surv. 3 (2006), 206–229 (electronic).
  • T. Imamura, T. Sasamoto. Dynamics of a tagged particle in the asymmetric exclusion process with the step initial condition. J. Stat. Phys. 128 (2007), no. 4, 799–846.
  • Kurt Johansson. Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 (2003), no. 1-2, 277–329.
  • Kurt Johansson. The arctic circle boundary and the Airy process. Ann. Probab. 33 (2005), no. 1, 1–30.
  • Kurt Johansson. Random matrices and determinantal processes. Mathematical Statistical Physics, Session LXXXIII: Lecture Notes of the Les Houches Summer School 2005 edited by A. Bovier, F. Dunlop, A. van Enter, F. den Hollander, and J. Dalibard. Elsevier Science, 2006, pp. 1–56.
  • Russell Lyons. Determinantal probability measures. Publ. Math. Inst. Hautes Études Sci. No. 98 (2003), 167–212.
  • Michael Prähofer, Herbert Spohn. Scale invariance of the PNG droplet and the Airy process. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. J. Statist. Phys. 108 (2002), no. 5-6, 1071–1106.
  • A. Rákos, G.M. Schütz. Bethe ansatz and current distribution for the TASEP with particle-dependent hopping rates. Markov Process. Related Fields 12 (2006), no. 2, 323–334.
  • Gunter M. Schütz. Exact solution of the master equation for the asymmetric exclusion process. J. Statist. Phys. 88 (1997), no. 1-2, 427–445.
  • A.B. Soshnikov. Determinantal random fields. Encyclopedia of Mathematical Physics edited by J.-P. Francoise, G. Naber, and T. S. Tsun. Elsevier, Oxford, 2006, pp. 47–53.
  • Herbert Spohn. Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals. Phys. A 369 (2006), no. 1, 71–99.