Quenched, annealed and functional large deviations for one-dimensional random walk in random environment

Abstract

Suppose that the integers are assigned random variables {ω i} (taking values in the unit interval), which serve as an environment. This environment defines a random walk {X n} (called a RWRE) which, when at i, moves one step to the right with probability ω i, and one step to the left with probability 1 − ω i. When the {ω i} sequence is i.i.d., Greven and den Hollander (1994) proved a large deviation principle for X n/n, conditional upon the environment, with deterministic rate function.We consider in this paper large deviations, both conditioned on the environment (quenched) and averaged on the environment (annealed), for the RWRE, in the ergodic environment case. The annealed rate function is the solution of a variational problem involving the quenched rate function and specific relative entropy. We also give a detailed qualitative description of the resulting rate functions. Our techniques differ from those of Greven and den Hollander, and allow us to present also a trajectorial (quenched) large deviation principle.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Alili, S.: Asymptotic behaviour for random walks in random environments, J. Appl. Prob. 36, 334–349 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Chung, K.L.: Markov chains with stationary transition probabilities, Berlin, Springer, 1960

    Google Scholar 

  3. 3.

    Dembo, A., Peres, Y., Zeitouni, O.: Tail estimates for one-dimensional random walk in random environment, Commun. Math. Phys. 181, 667–683 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, second edition, Springer, New York (1998)

    Google Scholar 

  5. 5.

    Dette, H., Fill, J.A., Pitman, J., Studden, W.J., Wall and Siegmund duality relations for birth and death chains with reflecting barrier, J. Theor. Prob. 10, 349–374 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time III, Comm. Pure Appl. Math. 29, 389–461 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time IV, Comm. Pure Appl. Math. 36, 183–212 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    Föllmer, H.: Random fields and diffusion processes. In: Ecole d’Eté de Saint Flour XV-XVII, Springer Lect. Notes in Math. 1362, 101–203 (1988)

    Google Scholar 

  9. 9.

    Gantert, N., Zeitouni, O.: Quenched sub-exponential tail estimates for one-dimensional random walk in random environment. Commun. Math. Phys. 194, 177–190 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Georgii, H.-O.: Gibbs Measures and Phase Transitions. De Gruyter, Berlin (1988)

    Google Scholar 

  11. 11.

    Greven, A., den Hollander, F.: Large deviations for a random walk in random environment, Ann. Probab. 22, 1381–1428 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    Hughes, B. D.: Random walks and random environments, Clarendon Press, Oxford (1996)

    Google Scholar 

  13. 13.

    Jones, W.B., Thron, W.J.: Continued fractions, Encyclopedia of Mathematics and its Applications, vol. 11, Addison-Wesley, 1980

    Google Scholar 

  14. 14.

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

    MATH  MathSciNet  Google Scholar 

  15. 15.

    Kozlov, M.V.: Random walk in a one-dimensional random medium, Th. Prob. and Appl. 18, 387–388 (1973)

    Article  MATH  Google Scholar 

  16. 16.

    Ledrappier, F.: Quelques propriétés des exposants charactéristiques, Springer, LNM 1097, 305–396 (1984)

    Google Scholar 

  17. 17.

    Pisztora, A., Povel, T.: Large deviation principle for random walk in a quenched environment in the low speed regime, Ann. Probab. 27, 1389–1413 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. 18.

    Pisztora, A., Povel, T., Zeitouni, O.: Precise large deviations estimates for one dimensional random walk in random environment, Probab. Theory Related Fields 113, 171–190 (1999)

    Article  MathSciNet  Google Scholar 

  19. 19.

    Révész, P.: Random walk in random and non-random environments, World Scientific, Singapore (1990)

    Google Scholar 

  20. 20.

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

    Article  MATH  Google Scholar 

  21. 21.

    Spitzer, F.: Principles of random walk, Springer, Berlin (1976)

    Google Scholar 

  22. 22.

    Sznitman, A.-S.: Slowdown and neutral pockets for a random walk in a random environment, Probab. Theory Related Fields 115, 287–323 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  23. 23.

    Wall, H.S.: Analytic theory of continued fractions, Van Nostrand, 1948

    Google Scholar 

  24. 24.

    Zerner, M.P.W.: Lyapunov exponents and quenched large deviations for multi-dimensional random walk in random environment, Ann. Probab. 26, 1446–1476 (1998)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Francis Comets.

Additional information

Research partially supported by CNRS, UMR7599. Part of this work was done while visiting the Dept. of Electrical Eng., Technion, Israel.

Research supported by the Swiss National Science foundation under grant 8220-046518, and partially supported by the DFG. Part of this work was done while visiting the Dept. of Electrical Eng., Technion, Israel.

Partially supported by a US-Israel BSF grant and by IHP.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Comets, F., Gantert, N. & Zeitouni, O. Quenched, annealed and functional large deviations for one-dimensional random walk in random environment. Probab Theory Relat Fields 118, 65–114 (2000). https://doi.org/10.1007/s004400000074

Download citation

Mathematics Subject Classification (2000)

  • 60J15
  • 60F10
  • 82C44
  • 60J80

Key words and phrases

  • Random walk in random environment
  • Large deviations