Annals of Probability

Quenched limits for transient, zero speed one-dimensional random walk in random environment

Jonathon Peterson and Ofer Zeitouni

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We consider a nearest-neighbor, one dimensional random walk {Xn}n≥0 in a random i.i.d. environment, in the regime where the walk is transient but with zero speed, so that Xn is of order ns for some s<1. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible: There exist sequences {nk} and {xk} depending on the environment only, such that Xnkxk=o(log nk)2 (a localized regime). On the other hand, there exist sequences {tm} and {sm} depending on the environment only, such that logsm/log tms<1 and Pω(Xtm/smx)→1/2 for all x>0 and →0 for x≤0 (a spread out regime).

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Ann. Probab., Volume 37, Number 1 (2009), 143-188.

First available in Project Euclid: 17 February 2009

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Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments
Secondary: 60F05: Central limit and other weak theorems 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50] 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Random walk random environment


Peterson, Jonathon; Zeitouni, Ofer. Quenched limits for transient, zero speed one-dimensional random walk in random environment. Ann. Probab. 37 (2009), no. 1, 143--188. doi:10.1214/08-AOP399.

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  • [1] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Applications of Mathematics (New York) 38. Springer, New York.
  • [2] Enriquez, N., Sabot, C. and Zindy, O. (2007). Limit laws for transient random walks in random environment on ℤ. Preprint. arXiv:math/0703660v3 [math.PR].
  • [3] Fitzsimmons, P. J. and Pitman, J. (1999). Kac’s moment formula and the Feynman–Kac formula for additive functionals of a Markov process. Stochastic Process. Appl. 79 117–134.
  • [4] Gantert, N. and Shi, Z. (2002). Many visits to a single site by a transient random walk in random environment. Stochastic Process. Appl. 99 159–176.
  • [5] Goldsheid, I. Y. (2007). Simple transient random walks in one-dimensional random environment: The central limit theorem. Probab. Theory Related Fields 139 41–64.
  • [6] Iglehart, D. L. (1972). Extreme values in the GI/G/1 queue. Ann. Math. Statist. 43 627–635.
  • [7] Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207–248.
  • [8] Kesten, H., Kozlov, M. V. and Spitzer, F. (1975). A limit law for random walk in a random environment. Compositio Math. 30 145–168.
  • [9] Kobus, M. (1995). Generalized Poisson distributions as limits of sums for arrays of dependent random vectors. J. Multivariate Anal. 52 199–244.
  • [10] Kozlov, S. M. and Molchanov, S. A. (1984). Conditions for the applicability of the central limit theorem to random walks on a lattice. Dokl. Akad. Nauk SSSR 278 531–534.
  • [11] Peterson, J. (2008). Ph.D. thesis. Awarded in 2008 by the University of Minnesota. Available at arXiv:0810.257v1 [math.PR].
  • [12] Rassoul-Agha, F. and Seppäläinen, T. (2006). Ballistic random walk in a random environment with a forbidden direction. ALEA Lat. Am. J. Probab. Math. Stat. 1 111–147 (electronic).
  • [13] Solomon, F. (1975). Random walks in a random environment. Ann. Probab. 3 1–31.
  • [14] Zeitouni, O. (2004). Random walks in random environment. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1837 189–312. Springer, Berlin.