### Abstract

We study once-reinforced biased random walk on Z^{d}. We prove that for sufficiently large bias, the speed v(β) is monotone decreasing in the reinforcement parameter β in the region [0,β_{0}], where β_{0} is a small parameter depending on the underlying bias. This result is analogous to results on Galton–Watson trees obtained by Collevecchio and the authors.

Original language | English |
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Title of host publication | Sojourns in Probability Theory and Statistical Physics - III |

Subtitle of host publication | Interacting Particle Systems and Random Walks, A Festschrift for Charles M. Newman |

Editors | Vladas Sidoravicius |

Place of Publication | Singapore |

Publisher | Springer Nature |

Pages | 255-273 |

Number of pages | 19 |

Volume | 300 |

ISBN (Electronic) | 9789811503023 |

ISBN (Print) | 9789811503016 |

DOIs | |

Publication status | E-pub ahead of print - 18 Oct 2019 |

Event | International Conference on Probability Theory and Statistical Physics, 2016 - Shanghai, China Duration: 25 Mar 2016 → 27 Mar 2016 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 300 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Conference

Conference | International Conference on Probability Theory and Statistical Physics, 2016 |
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Country | China |

City | Shanghai |

Period | 25/03/16 → 27/03/16 |

### Keywords

- Coupling
- Large bias
- Once-reinforced random walk
- Reinforced random walk

### ASJC Scopus subject areas

- Mathematics(all)

## Cite this

Holmes, M., & Kious, D. (2019). A Monotonicity Property for Once Reinforced Biased Random Walk on Z

^{d}. In V. Sidoravicius (Ed.),*Sojourns in Probability Theory and Statistical Physics - III: Interacting Particle Systems and Random Walks, A Festschrift for Charles M. Newman*(Vol. 300, pp. 255-273). (Springer Proceedings in Mathematics and Statistics; Vol. 300). Springer Nature. https://doi.org/10.1007/978-981-15-0302-3_10