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Quenched, annealed and functional large deviations for one-dimensional random walk in random environment
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  • Published: 05 September 2000

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

  • Francis Comets1,
  • Nina Gantert2 &
  • Ofer Zeitouni3 

Probability Theory and Related Fields volume 118, pages 65–114 (2000)Cite this article

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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.

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Author information

Authors and Affiliations

  1. UFR de Mathématiques, Case 7012, UniversitéParis 7, 2, place Jussieu, F-75251, Paris Cedex 05, France

    Francis Comets

  2. Department of Mathematics, TU Berlin, Strasse des 17. Juni 135, D-10623, Berlin, Germany

    Nina Gantert

  3. Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa o[32000, Israel

    Ofer Zeitouni

Authors
  1. Francis Comets
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  2. Nina Gantert
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  3. Ofer Zeitouni
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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.

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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

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  • Received: 21 July 1998

  • Revised: 27 October 1999

  • Published: 05 September 2000

  • Issue Date: September 2000

  • DOI: https://doi.org/10.1007/s004400000074

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Mathematics Subject Classification (2000)

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

Key words and phrases

  • Random walk in random environment
  • Large deviations
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