Consider the massless free field on the d-dimensional lattice ℤ d , d≥3; that is the centered Gaussian field on ℝ ℤ d with covariances given by the Green function of the simple random walk on ℤ d . We show that the probability, that all the spins are positive in a box of volume N d , decays exponentially at a rate of order N d-2 logN and compute explicitly the corresponding constant in terms of the capacity of the unit cube. The result is extended to a class of transient random walks with transition functions in the domain of the normal and α-stable law.

Bolthausen, E; Deuschel, J-D; Zeitouni, O (1995). *Entropic repulsion of the lattice free field.* Communications in Mathematical Physics, 170(2):417-443.

## Abstract

Consider the massless free field on the d-dimensional lattice ℤ d , d≥3; that is the centered Gaussian field on ℝ ℤ d with covariances given by the Green function of the simple random walk on ℤ d . We show that the probability, that all the spins are positive in a box of volume N d , decays exponentially at a rate of order N d-2 logN and compute explicitly the corresponding constant in terms of the capacity of the unit cube. The result is extended to a class of transient random walks with transition functions in the domain of the normal and α-stable law.

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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 |

Uncontrolled Keywords: | large deviations; Green function; random walk |

Language: | English |

Date: | 1995 |

Deposited On: | 27 Apr 2010 14:08 |

Last Modified: | 05 Apr 2016 13:27 |

Publisher: | Springer |

ISSN: | 0010-3616 |

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

Free access at: | Related URL. An embargo period may apply. |

Publisher DOI: | https://doi.org/10.1007/BF02108336 |

Related URLs: | http://projecteuclid.org/euclid.cmp/1104273128 (Organisation) http://www.zentralblatt-math.org/NEW/zmath/search/?q=an%3A0821.60040 http://www.ams.org/mathscinet-getitem?mr=1334403 https://www.zora.uzh.ch/22069 |

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