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.

Bolthausen, E; Goldsheid, I (2008). *Lingering random walks in random environment on a strip.* Communications in Mathematical Physics, 278(1):253-288.

## 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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## Additional indexing

Item Type: | Journal Article, refereed, original work |
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Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | 2008 |

Deposited On: | 07 Jan 2009 16:33 |

Last Modified: | 05 Apr 2016 12:39 |

Publisher: | Springer |

ISSN: | 0010-3616 |

Additional Information: | The original publication is available at www.springerlink.com |

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

Related URLs: | http://arxiv.org/abs/0710.5854 |

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