## Abstract

We introduce a Gibbs measure on nearest-neighbour paths of length t in the Euclidean d-dimensional lattice, where each path is penalised by a factor proportional to the size of its boundary and an inverse temperature β. We prove that, for all β > 0, the random walk condensates to a set of diameter (t/β) ^{1} / ^{3} in dimension d = 2, up to a multiplicative constant. In all dimensions d ≥ 3, we also prove that the volume is bounded above by (t/β) ^{d/(d} +1 ^{)} and the diameter is bounded below by (t/β) ^{1} /(d+1) Similar results hold for a random walk conditioned to have local time greater than β everywhere in its range when β is larger than . some explicit constant, which in dimension two is the logarithm of the connective constant.

Original language | English |
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Pages (from-to) | 835-861 |

Number of pages | 27 |

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

Volume | 55 |

Issue number | 2 |

DOIs | |

State | Published - 1 May 2019 |

## Keywords

- Condensation
- Donsker–Varadhan principle
- Gibbs measure
- Large deviations
- Self-attractive random walk
- Wulff crystal

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty