## Abstract

It is well known that the distribution of simple random walks on ℤ conditioned on returning to the origin after 2n steps does not depend on p = P(S_{1} = 1), the probability of moving to the right. Moreover, conditioned on {S_{2n} = 0} the maximal displacement max _{k≤2n} |S_{k}| converges in distribution when scaled by √n (diffusive scaling). We consider the analogous problem for transient random walks in random environments on ℤ.We show that under the quenched law P_{ω} (conditioned on the environment ω), the maximal displacement of the random walk when conditioned to return to the origin at time 2n is no longer necessarily of the order √n. If the environment is nestling (both positive and negative local drifts exist) then the maximal displacement conditioned on returning to the origin at time 2n is of order n^{κ/(κ+1)}, where the constantκ >0 depends on the law on environments. On the other hand, if the environment is marginally nestling or non-nestling (only non-negative local drifts) then the maximal displacement conditioned on returning to the origin at time 2n is at least n^{1-ε} and at most n/(ln n)^{2-ε} for any ε > 0. As a consequence of our proofs, we obtain precise rates of decay for P _{ω}(X_{2n} = 0). In particular, for certain non-nestling environments we show that P_{ω}(X_{2n} = 0) = exp{-Cn-C″n/(ln n)^{2} + o(n/(ln n)^{2})} with explicit constants C,C″ > 0.

Original language | English |
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Pages (from-to) | 663-678 |

Number of pages | 16 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 47 |

Issue number | 3 |

DOIs | |

State | Published - Aug 2011 |

Externally published | Yes |

## Keywords

- Moderate deviations
- Random walk in random environment