The Annals of Probability

Current fluctuations of a system of one-dimensional random walks in random environment

Jonathon Peterson and Timo Seppäläinen

Full-text: Open access


We study the current of particles that move independently in a common static random environment on the one-dimensional integer lattice. A two-level fluctuation picture appears. On the central limit scale the quenched mean of the current process converges to a Brownian motion. On a smaller scale the current process centered at its quenched mean converges to a mixture of Gaussian processes. These Gaussian processes are similar to those arising from classical random walks, but the environment makes itself felt through an additional Brownian random shift in the spatial argument of the limiting current process.

Article information

Ann. Probab., Volume 38, Number 6 (2010), 2258-2294.

First available in Project Euclid: 24 September 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Random walk in random environment current fluctuations central limit theorem


Peterson, Jonathon; Seppäläinen, Timo. Current fluctuations of a system of one-dimensional random walks in random environment. Ann. Probab. 38 (2010), no. 6, 2258--2294. doi:10.1214/10-AOP537.

Export citation


  • [1] Arratia, R. (1983). The motion of a tagged particle in the simple symmetric exclusion system on Z. Ann. Probab. 11 362–373.
  • [2] Balázs, M. and Seppäläinen, T. (2010). Order of current variance and diffusivity in the asymmetric simple exclusion process. Ann. of Math. 171 1237–1265.
  • [3] Dürr, D., Goldstein, S. and Lebowitz, J. L. (1985). Asymptotics of particle trajectories in infinite one-dimensional systems with collisions. Comm. Pure Appl. Math. 38 573–597.
  • [4] Durrett, R. (1996). Probability: Theory and Examples, 2nd ed. Duxbury Press, Belmont, CA.
  • [5] 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.
  • [6] Goldsheid, I. Y. (2007). Simple transient random walks in one-dimensional random environment: The central limit theorem. Probab. Theory Related Fields 139 41–64.
  • [7] Jara, M. (2009). Current and density fluctuations for interacting particle systems with anomalous diffusive behavior. Available at arXiv:0901.0229.
  • [8] Jara, M. D. and Landim, C. (2006). Nonequilibrium central limit theorem for a tagged particle in symmetric simple exclusion. Ann. Inst. H. Poincaré Probab. Statist. 42 567–577.
  • [9] Jara, M. D. and Landim, C. (2008). Quenched non-equilibrium central limit theorem for a tagged particle in the exclusion process with bond disorder. Ann. Inst. H. Poincaré Probab. Statist. 44 341–361.
  • [10] Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476.
  • [11] Kesten, H., Kozlov, M. V. and Spitzer, F. (1975). A limit law for random walk in a random environment. Compositio Math. 30 145–168.
  • [12] Kumar, R. (2008). Space–time current process for independent random walks in one dimension. ALEA Lat. Am. J. Probab. Math. Stat. 4 307–336.
  • [13] Mayer-Wolf, E., Roitershtein, A. and Zeitouni, O. (2004). Limit theorems for one-dimensional transient random walks in Markov environments. Ann. Inst. H. Poincaré Probab. Statist. 40 635–659.
  • [14] Peligrad, M. and Sethuraman, S. (2008). On fractional Brownian motion limits in one dimensional nearest-neighbor symmetric simple exclusion. ALEA Lat. Am. J. Probab. Math. Stat. 4 245–255.
  • [15] Peterson, J. (2008). Limiting distributions and large deviations for random walks in random environments. Ph.D. thesis, Univ. Minnesota. Available at arXiv:0810.0257v1.
  • [16] Peterson, J. (2009). Quenched limits for transient, ballistic, sub-Gaussian one-dimensional random walk in random environment. Ann. Inst. H. Poincaré Probab. Statist. 45 685–709.
  • [17] Peterson, J. (2009). Systems of one-dimensional random walks in a common random environment. Preprint. Available at arXiv:0907.3680v1.
  • [18] Peterson, J. and Zeitouni, O. (2009). Quenched limits for transient, zero speed one-dimensional random walk in random environment. Ann. Probab. 37 143–188.
  • [19] Quastel, J. and Valko, B. (2007). t1/3 superdiffusivity of finite-range asymmetric exclusion processes on ℤ. Comm. Math. Phys. 273 379–394.
  • [20] Seppäläinen, T. (2005). Second-order fluctuations and current across characteristic for a one-dimensional growth model of independent random walks. Ann. Probab. 33 759–797.
  • [21] Solomon, F. (1975). Random walks in a random environment. Ann. Probab. 3 1–31.
  • [22] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École D’été de Probabilités de Saint-Flour, XIV—1984. Lecture Notes in Math. 1180 265–439. Springer, Berlin.
  • [23] Zeitouni, O. (2004). Random walks in random environment. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1837 189–312. Springer, Berlin.