In this survey paper we mainly discuss the results contained in two of our recent articles [2] and [5]. Let {Xt}t≥0 be a continuous-time, symmetric, nearest-neighbour random walk on Zd. For every T > 0 we define the transformed

path measure dPT = (1/ZT ) exp(HT ) dP, where P is the original one and ZT is the appropriate normalizing constant. The Hamiltonian HT imparts the self-attracting

interaction of the paths up to time T. We consider the case where HT is given by a potential function V on Zd with finite support, and the case HT = −NT , where NT denotes the number of points visited by the random walk up to time T. In both cases the typical paths under PT as T →∞ clump together much more than those of the free random walk and give rise to localization phenomena.

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Bolthausen, E; Schmock, U (1995). *On self-attracting random walks.* In: Cranston, M C; Pinsky, M A. Stochastic analysis (Ithaca, NY, 1993). Providence, RI: American Mathematical Society, 23-44.

## Abstract

In this survey paper we mainly discuss the results contained in two of our recent articles [2] and [5]. Let {Xt}t≥0 be a continuous-time, symmetric, nearest-neighbour random walk on Zd. For every T > 0 we define the transformed

path measure dPT = (1/ZT ) exp(HT ) dP, where P is the original one and ZT is the appropriate normalizing constant. The Hamiltonian HT imparts the self-attracting

interaction of the paths up to time T. We consider the case where HT is given by a potential function V on Zd with finite support, and the case HT = −NT , where NT denotes the number of points visited by the random walk up to time T. In both cases the typical paths under PT as T →∞ clump together much more than those of the free random walk and give rise to localization phenomena.

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## Additional indexing

Item Type: | Book Section, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | nearest-neighbor random walk; self-attracting interaction; localization phenomena |

Language: | English |

Date: | 1995 |

Deposited On: | 20 May 2010 12:53 |

Last Modified: | 05 Apr 2016 13:27 |

Publisher: | American Mathematical Society |

Series Name: | Proceedings of Symposia in Pure Mathematics |

Number: | 57 |

ISSN: | 0082-0717 |

ISBN: | 0-8218-0289-5 |

Additional Information: | First published in [On self-attracting random walks. Stochastic analysis (Ithaca, NY, 1993), 23--44, Proc. Sympos. Pure Math., 57], published by the American Mathematical Society |

Official URL: | http://www.ams.org/bookstore?fn=20&arg1=pspumseries&ikey=PSPUM-57 |

Related URLs: | http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0829.60021 http://www.fam.tuwien.ac.at/~schmock/Attracting_walks.html |

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