Large deviation principle for random walk in a quenched random environment in the flow speed regime.

*(English)*Zbl 0964.60056The authors consider a one-dimensional random walk in an i.i.d. random environment. The random environment has \`\` positive and zero drifts\'\': depending on the site where the particle is, it moves either to the left or to the right with equal probabilites (\`\` fair\'\' sites), or it has a drift to the right (\`\` biased\'\' sites). In such an environment, the random walk has almost surely a positive, deterministic speed \(v_\alpha\). For almost all environments, the probabilities that the average position \(X_n/n\) is smaller than \(v\), for some \(v < v_\alpha\), decays for \(n\) going to infinity. The order of this decay is \(\exp(-n/(\log n)^2)\), see the reviewer and O. Zeitouni [Commun. Math. Phys. 194, No. 1, 177-190 (1998)]. The authors extend this statement and provide the rate function on \((0, v_\alpha)\), thereby proving a large deviation principle of order \(n/(\log n)^2\) for \(X_n/n\) in this regime. The main technique used in the proof of the upper bound is a coarse graining procedure where the environment is split up in blocks, and the blocks are classified to be fair or biased, according to the proportion of the fair sites in the block. A similar coarse graining technique had been used in the context of Brownian motion in a Poissonian potential by one of the authors, see T. Povel [Ann. Probab. 25, No. 4, 1735-1773 (1997; Zbl 0911.60014)].

Reviewer: Nina Gantert (Berlin)

##### MSC:

60G50 | Sums of independent random variables; random walks |

60F10 | Large deviations |

82C44 | Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics |

60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |

PDF
BibTeX
XML
Cite

\textit{A. Pisztora} and \textit{T. Povel}, Ann. Probab. 27, No. 3, 1389--1413 (1999; Zbl 0964.60056)

Full Text:
DOI

**OpenURL**

##### References:

[1] | AZUMA, K. 1967. Weighted sums of certain dependent random variables. Tohoku Math. J. 19 357 367. · Zbl 0178.21103 |

[2] | CHUNG, K. L. 1967. Markov Chains with Stationary Transition Probabilities. Springer, Berlin. · Zbl 0146.38401 |

[3] | DEMBO, A., PERES, Y. and ZEITOUNI, O. 1996. Tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 181 667 683. · Zbl 0868.60058 |

[4] | GANTERT, N. and ZEITOUNI, O. 1998. Quenched sub-exponential tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 194 177 190. · Zbl 0982.60037 |

[5] | GANTERT, N. and ZEITOUNI, O. 1998. Recent results on large deviations for one-dimensional random walk in random environment. · Zbl 0910.60013 |

[6] | GREVEN, A. and DEN HOLLANDER, F. 1994. “Large deviations for a random walk in random environment.” Ann. Probab. 22 1381 1428. · Zbl 0820.60054 |

[7] | PISZTORA, A., POVEL, T. and ZEITOUNI, O. 1997. Precise large deviation estimates for one-dimensional random walk in random environment. Probab. Theory Related Fields. · Zbl 0922.60059 |

[8] | POVEL, T. 1997. Critical large deviations of one-dimensional annealed Brownian motion in a Poissonian potential. Ann. Probab. 25 1735 1773. · Zbl 0911.60014 |

[9] | SOLOMON, F. 1975. Random walks in random environment. Ann. Probab. 3 1 31. · Zbl 0305.60029 |

[10] | SZNITMAN, A.-S. 1995. Quenched critical large deviations for Brownian motion in a Poissonian potential. J. Funct. Anal. 131 54 77. · Zbl 0853.60027 |

[11] | PITTSBURGH, PENNSYLVANIA 15213 SWITZERLAND E-MAIL: pisztora@andrew.cmu.edu E-MAIL: tobias.povel@zurichre.com |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.