### Abstract

Friendly walkers is a stochastic model obtained from independent one-dimensional simple random walks {S_{j}^{k}}_{j≥0}, k = 1, 2,..., d by introducing "non-crossing condition": S_{j}^{1} ≤ S_{j}^{2} ≤ ... ≤ S_{j}^{d}, j = 1,2, ..., n and "reward for collisions" characterized by parameters β_{2}, ..., β_{d} ≥ 0. Here, the reward for collisions is described as follows. If, at a given time n, a site in ℤ is occupied by exactly m ≥ 2 walkers, then the site increases the probabilistic weight for the walkers by multiplicative factor exp(β_{m}) ≥ 1. We study the localization transition of this model in terms of the positivity of the free energy and describe the location and the shape of the critical surface in the (d - 1)-dimensional space for the parameters (β_{2},..., β_{d}).

Original language | English |
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Pages (from-to) | 593-608 |

Number of pages | 16 |

Journal | Probability Theory and Related Fields |

Volume | 125 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2003 Apr 1 |

Externally published | Yes |

### Keywords

- Lattice animals
- Phase transitions
- Polymers
- Random surfaces
- Random walks

### ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Probability Theory and Related Fields*,

*125*(4), 593-608. https://doi.org/10.1007/s00440-002-0253-z