Consider a one-dimensional walk (Sk)k having steps of bounded size, and weight the probability of the path with some factor 1−α∈(0,1) for every single self-intersection up to time n. We prove that Sn/n converges towards some deterministic number called the effective drift of the self-repellent walk. Furthermore, this drift is shown to tend to the basic drift as α tends to 0 and, as α tends to 1, to the self-avoiding walk's drift which was introduced in an earlier paper of ours [Probab. Theory Related Fields 96 (1993), no. 4, 521--543]. The main tool of the present paper is a representation of the sequence of the local times as a functional of a certain Markov process.

König, W (1994). *The drift of a one-dimensional self-repellent random walk with bounded increments.* Probability Theory and Related Fields, 100(4):513-544.

## Abstract

Consider a one-dimensional walk (Sk)k having steps of bounded size, and weight the probability of the path with some factor 1−α∈(0,1) for every single self-intersection up to time n. We prove that Sn/n converges towards some deterministic number called the effective drift of the self-repellent walk. Furthermore, this drift is shown to tend to the basic drift as α tends to 0 and, as α tends to 1, to the self-avoiding walk's drift which was introduced in an earlier paper of ours [Probab. Theory Related Fields 96 (1993), no. 4, 521--543]. The main tool of the present paper is a representation of the sequence of the local times as a functional of a certain Markov process.

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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: | 1994 |

Deposited On: | 29 Nov 2010 16:29 |

Last Modified: | 05 Apr 2016 13:28 |

Publisher: | Springer |

ISSN: | 0178-8051 |

Publisher DOI: | https://doi.org/10.1007/BF01268992 |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=1305785 http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0810.60095 |

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