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Lingering Random Walks in Random Environment on a Strip

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Abstract

We consider a recurrent random walk (RW) in random environment (RE) on a strip. We prove that if the RE is i. i. d. and its distribution is not supported by an algebraic subsurface in the space of parameters defining the RE then the RW exhibits the (log t)2 asymptotic behaviour. The exceptional algebraic subsurface is described by an explicit system of algebraic equations.

One-dimensional walks with bounded jumps in a RE are treated as a particular case of the strip model. If the one dimensional RE is i. i. d., then our approach leads to a complete and constructive classification of possible types of asymptotic behaviour of recurrent random walks. Namely, the RW exhibits the (log t)2 asymptotic behaviour if the distribution of the RE is not supported by a hyperplane in the space of parameters which shall be explicitly described. And if the support of the RE belongs to this hyperplane then the corresponding RW is a martingale and its asymptotic behaviour is governed by the Central Limit Theorem.

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References

  1. Bolthausen, E., Goldsheid, I.: Recurrence and transience of random walks in random environments on a strip. Commun. Math. Phys. 214, 429–447 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Brémont, J.: On some random walks on Z in random medium. Ann. Probab. 30, 1266–1312 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brémont, J.: Behavior of random walks on Z in Gibbsian medium. C. R. Acad. Sci. Série 1 Math. 338(11), 895–898 (2004)

    MATH  Google Scholar 

  4. Brown, B.M.: Martingale Central Limit Theorems. Ann. Math. Statist. 42, 59–66 (1971)

    Article  MathSciNet  Google Scholar 

  5. Furstenberg, H., Kesten, H.: Products of random matrices. Ann. Math. Statist. 31, 457–469 (1960)

    Article  MathSciNet  Google Scholar 

  6. Goldsheid, I.: Linear and Sub-linear Growth and the CLT for Hitting Times of a Random Walk in Random Environment on a Strip. Probability Theory and Related Fields, appeared on line in August, 2007, DOI: 10.1007/s00440-007-0091-0

  7. Golosov, A.: Localization of random walks in one-dimensional random environments. Commun. Math. Phys. 92, 491–506 (1984)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. Golosov, A.: On the limit distributions for a random walk in a critical one-dimensional random environment. Usp. Mat. Nauk 41(2), 189–190 (1986)

    MathSciNet  Google Scholar 

  9. Gordin, M.I.: The Central Limit Theorem for stationary processes. Soviet Math. Dokl. 10, 1174–1176 (1969)

    MATH  Google Scholar 

  10. Gordin, M.I., Lifshits, B.A.: The Central Limit Theorem for stationary Markov processes. Sov. Math. Dokl. 19(2), 392–394 (1978)

    MATH  Google Scholar 

  11. Gordin, M.I., Lifshits, B.A.: The Invariance principle for stationary Markov processes. “Teorija verojatnostej i ejo primenenija” 1978, issue 4, pp. 865-866 (in Russian)

  12. Hall, P., Heyde, C.C.: Martingale limit theory and its application. Academic Press, New York (1980)

    MATH  Google Scholar 

  13. Kesten, H.: The limit distribution of Sinai’s random walk in a random environment. Physica A 138, 299–309 (1986)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Kesten, H., Kozlov, M.V., Spitzer, F.: Limit law for random walk in a random environment. Comp. Math. 30, 145–168 (1975)

    MATH  MathSciNet  Google Scholar 

  15. Key, E.: Recurrence and transience criteria for random walk in a random environment. Ann. Prob. 12, 529–560 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lawler, G.: Weak convergence of a random walks in a random environment. Commun. Math. Phys. 87, 81–87 (1982)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. Letchikov, A.V.: Localization of one-dimensional random walks in random environment. Soviet Scientific Reviews Section C: Mathematical Physics Reviews. Chur. Switzerland: Harwood Academic Publishers, 1989, pp. 173–220

  18. Sinai, Ya.G.: The limiting behavior of a one-dimensional random walk in a random medium. Theory Prob. Appl. 27, 256–268 (1982)

    Article  MathSciNet  Google Scholar 

  19. Solomon, F.: Random walks in a random environment. Ann. Prob. 3, 1–31 (1975)

    Article  MATH  Google Scholar 

  20. Zeitouni, O.: Random walks in random environment, XXXI Summer school in Probability, St. Flour (2001). Lecture notes in Math. 1837, Berlin:Springer, 2004, pp. 193–312

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Authors and Affiliations

  1. Universität Zürich, Institut für Mathematik, Winterthurerstrasse 190, CH-8057, Zürich, Switzerland

    Erwin Bolthausen

  2. School of Mathematical Sciences, Queen Mary and Westfield College, University of London, London, E1 4NS, UK

    Ilya Goldsheid

Authors
  1. Erwin Bolthausen
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  2. Ilya Goldsheid
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Correspondence to Erwin Bolthausen.

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Communicated by M. Aizenman

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Bolthausen, E., Goldsheid, I. Lingering Random Walks in Random Environment on a Strip. Commun. Math. Phys. 278, 253–288 (2008). https://doi.org/10.1007/s00220-007-0390-4

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  • DOI: https://doi.org/10.1007/s00220-007-0390-4

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