The Annals of Probability

On new examples of ballistic random walks in random environment

Alain-Sol Sznitman

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Abstract

In this article we show that random walks in random environment on $\mathbb{Z}^d$, $d \ge3$, with transition probabilities which are $\varepsilon$-perturbations of the simple random walk and such that the expectation of the local drift has size bigger than $\varepsilon^\rho $, with $\rho< \frac{5}{2}$, when $d=3$, $\rho< 3$, when $d \ge4$, fulfill the condition (T$^\prime$) introduced by Sznitman [Prob. Theory Related Fields (2002) 122 509-544], when $\varepsilon$ is small. As a result these walks satisfy a law of large numbers with nondegenerate limiting velocity, a central limit theorem and several large deviation controls. In particular, this provides examples of ballistic random walks in random environment which do not satisfy Kalikow's condition in the terminology of Sznitman and Zerner [Ann. Probab. (1999) 27 1851-1869]. An important tool in this work is the effective criterion of Sznitman.

Article information

Source
Ann. Probab., Volume 31, Number 1 (2003), 285-322.

Dates
First available in Project Euclid: 26 February 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1046294312

Digital Object Identifier
doi:10.1214/aop/1046294312

Mathematical Reviews number (MathSciNet)
MR1959794

Zentralblatt MATH identifier
1017.60104

Subjects
Primary: 60K37: Processes in random environments 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Keywords
Random walk in random environment ballistic behavior small perturbations of simple random walk

Citation

Sznitman, Alain-Sol. On new examples of ballistic random walks in random environment. Ann. Probab. 31 (2003), no. 1, 285--322. doi:10.1214/aop/1046294312. https://projecteuclid.org/euclid.aop/1046294312


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References

  • [1] ALON, N., SPENCER, J. and ERDÖS, P. (1992). The Probabilistic Method. Wiley, New York.
  • [2] ANSHELEVICH, V. V., KHANIN, K. M. and SINAI, YA. G. (1982). Sy mmetric random walks in random environments. Comm. Math. Phy s. 85 449-470.
  • [3] BOLTHAUSEN, E. and SZNITMAN, A. S. (2002). Ten Lectures on Random Media. Birkhäuser, Basel.
  • [4] BOLTHAUSEN, E., SZNITMAN, A. S. and ZEITOUNI, O. (2002). Cut points and diffusive random walks in random environment. Ann. Inst. H. Poincaré Probab. Statist. To appear.
  • [5] BRICMONT, J. and KUPIAINEN, A. (1991). Random walks in asy mmetric random environments. Comm. Math. Phy s. 142 345-420.
  • [6] HUGHES, B. D. (1996). Random Walks and Random Environments 2. Clarendon Press, Oxford.
  • [7] KALIKOW, S. A. (1981). Generalized random walk in a random environment. Ann. Probab. 9 753-768.
  • [8] KHAS'MINSKII, R. Z. (1959). On positive solutions of the equation Au + V u = 0. Theoret. Probab. Appl. 4 309-318.
  • [9] KOZLOV, S. M. (1985). The method of averaging and walks in inhomogeneous environments. Russian Math. Survey s 40 73-145.
  • [10] LAWLER, G. F. (1982). Weak convergence of a random walk in a random environment. Comm. Math. Phy s. 87 81-87.
  • [11] LAWLER, G. F. (1991). Intersection of Random Walks. Birkhäuser, Basel.
  • [12] MOLCHANOV, S. A. (1992). Lectures on Random Media. Ecole d'eté de Probabilités de St. Flour XXII. Lecture Notes in Math. 1581. Springer, Berlin.
  • [13] OLLA, O. (1994). Homogenization of Diffusion Processes in Random Fields. Ecole Poly technique, Palaiseau.
  • [14] PAPANICOLAOU, G. and VARADHAN, S. R. S. (1981). Boundary value problems with rapidly oscillating random coefficients. In Random Fields (J. Fritz and D. Szasz, eds.) 835-873. North-Holland, Amsterdam.
  • [15] PAPANICOLAOU, G. and VARADHAN, S. R. S. (1982). Diffusion with Random Coefficients. Statistics and Probability: Essay s in Honor of C.R. Rao (G. Kallianpur, P. R. Krishnajah and J. K. Gosh, eds.) 547-552. North-Holland, Amsterdam.
  • [16] REED, M. and SIMON, B. (1975). Methods of Modern Mathematical physics 2. Academic Press, New York.
  • [17] SOLOMON, F. (1975). Random walk in random environment. Ann. Probab. 3 1-31.
  • [18] SZNITMAN, A. S. (1999). Slowdown and neutral pockets for a random walk in random environment. Probab. Theory Related Fields 115 287-323.
  • [19] SZNITMAN, A. S. (2000). Slowdown estimates and central limit theorem for random walks in random environment. J. Eur. Math. Soc. 2 93-143.
  • [20] SZNITMAN, A. S. (2001). On a class of transient random walks in random environment. Ann. Probab. 29 723-764.
  • [21] SZNITMAN, A. S. (2002). An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Related Fields 122 509-544.
  • [22] SZNITMAN, A. S. and ZERNER, M. P. W. (1999). A law of large numbers for random walks in random environment. Ann. Probab. 27 1851-1869.
  • [23] ZEITOUNI, O. (2001). Notes of Saint Flour lectures 2001. Preprint. Available at wwwee.technion.ac.il/ zeitouni/ps/notes1.ps.