Electronic Journal of Probability

Random walk driven by simple exclusion process

François Huveneers and François Simenhaus

Full-text: Open access


We prove strong law of large numbers and an annealed invariance principle for a random walk in a one-dimensional dynamic random environment evolving as the simple exclusion process with jump parameter $\gamma$.  First we establish that, if the asymptotic velocity of the walker is non-zero in the limiting case "$\gamma = \infty$" where the environment gets fully refreshed between each step, then, for $\gamma$ large enough, the walker still has a non-zero asymptotic velocity in the same direction.  Second we establish that if the walker is transient in the limiting case $\gamma = 0$, then, for $\gamma$ small enough but positive, the walker has a non-zero asymptotic velocity in the direction of the transience. These two limiting velocities can sometimes be of opposite sign. In all cases, we show that fluctuations are normal.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 105, 42 pp.

Accepted: 9 October 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments
Secondary: 60F17: Functional limit theorems; invariance principles

Random walk in dynamic random environment limit theorem renormalization renewal times

This work is licensed under aCreative Commons Attribution 3.0 License.


Huveneers, François; Simenhaus, François. Random walk driven by simple exclusion process. Electron. J. Probab. 20 (2015), paper no. 105, 42 pp. doi:10.1214/EJP.v20-3906. https://projecteuclid.org/euclid.ejp/1465067211

Export citation


  • Arratia, Richard. Symmetric exclusion processes: a comparison inequality and a large deviation result. Ann. Probab. 13 (1985), no. 1, 53–61.
  • L. Avena: Random Walks in Dynamic Random Environments. PhD thesis, 2010, www.catalogus.leidenuniv.nl
  • Avena, L.; den Hollander, F.; Redig, F. Law of large numbers for a class of random walks in dynamic random environments. Electron. J. Probab. 16 (2011), no. 21, 587–617.
  • Avena, Luca; dos Santos, Renato Soares; Völlering, Florian. Transient random walk in symmetric exclusion: limit theorems and an Einstein relation. ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013), no. 2, 693–709.
  • Avena, L.; Thomann, P. Continuity and anomalous fluctuations in random walks in dynamic random environments: numerics, phase diagrams and conjectures. J. Stat. Phys. 147 (2012), no. 6, 1041–1067.
  • Basu, Urna; Maes, Christian. Mobility transition in a dynamic environment. J. Phys. A 47 (2014), no. 25, 255003, 15 pp.
  • O. Bénichou et al.: Geometry-Induced Superdiffusion in Driven Crowded Systems. Physical Review Letters 111, (2013), 260601.
  • Bricmont, J.; Kupiainen, A. Random walks in asymmetric random environments. Comm. Math. Phys. 142 (1991), no. 2, 345–420.
  • J. Bérard, A. F. Ramirez: Fluctuations of the front in a one dimensional model for the spread of an infection. ARXIV1210.6781, 2012.
  • Bricmont, Jean; Kupiainen, Antti. Random walks in space time mixing environments. J. Stat. Phys. 134 (2009), no. 5-6, 979–1004.
  • Comets, Francis; Zeitouni, Ofer. A law of large numbers for random walks in random mixing environments. Ann. Probab. 32 (2004), no. 1B, 880–914.
  • den Hollander, Frank; Kesten, Harry; Sidoravicius, Vladas. Random walk in a high density dynamic random environment. Indag. Math. (N.S.) 25 (2014), no. 4, 785–799.
  • Dolgopyat, Dmitry; Keller, Gerhard; Liverani, Carlangelo. Random walk in Markovian environment. Ann. Probab. 36 (2008), no. 5, 1676–1710.
  • dos Santos, Renato Soares. Non-trivial linear bounds for a random walk driven by a simple symmetric exclusion process. Electron. J. Probab. 19 (2014), no. 49, 18 pp.
  • Harris, T. E. Additive set-valued Markov processes and graphical methods. Ann. Probability 6 (1978), no. 3, 355–378.
  • M. Hil' ario, F. den Hollander, V. Sidoravicius, R. S. dos Santos and A. Teixeira: Random Walk on Random Walks. ARXIV1401.4498, 2014.
  • Kalikow, Steven A. Generalized random walk in a random environment. Ann. Probab. 9 (1981), no. 5, 753–768.
  • Kesten, H.; Kozlov, M. V.; Spitzer, F. A limit law for random walk in a random environment. Compositio Math. 30 (1975), 145–168.
  • Liggett, Thomas M. Interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 276. Springer-Verlag, New York, 1985. xv+488 pp. ISBN: 0-387-96069-4.
  • Rassoul-Agha, Firas. The point of view of the particle on the law of large numbers for random walks in a mixing random environment. Ann. Probab. 31 (2003), no. 3, 1441–1463.
  • Redig, Frank; V0llering, Florian. Random walks in dynamic random environments: a transference principle. Ann. Probab. 41 (2013), no. 5, 3157–3180.
  • Sznitman, Alain-Sol. Slowdown estimates and central limit theorem for random walks in random environment. J. Eur. Math. Soc. (JEMS) 2 (2000), no. 2, 93–143.
  • Sznitman, Alain-Sol; Zerner, Martin. A law of large numbers for random walks in random environment. Ann. Probab. 27 (1999), no. 4, 1851–1869.
  • Zeitouni, Ofer. Random walks in random environment. Lectures on probability theory and statistics, 189–312, Lecture Notes in Math., 1837, Springer, Berlin, 2004.