## Abstract

We determine the geometrical properties of the most probable paths at finite temperatures T, between two points separated by a distance r, in one-dimensional lattices with positive energies of interaction ε_{i} associated with bond i. The most probable path-length t_{mp} in a homogeneous medium (ε_{i} = ε, for all i) is found to undergo a phase transition, from an optimal-like form (t_{mp} ∼ r) at low temperatures to a random walk form (t_{mp} ∼ r^{2}) near the critical temperature T_{c} = ε/ln 2. At T > T_{c} the most probable path-length diverges, discontinuously, for all finite endpoint separations greater than a particular value r*(T). In disordered lattices, with ε_{i} homogeneously distributed between ε-δ/2 and ε + δ/2, the random walk phase is absent, but a phase transition to diverging t_{mp} still takes place. Different disorder configurations have different transition points. A way to characterize the whole ensemble of disorder, for a given distribution, is suggested.

Original language | English |
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Pages (from-to) | 401-410 |

Number of pages | 10 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 297 |

Issue number | 3-4 |

DOIs | |

State | Published - 15 Aug 2001 |

### Bibliographical note

Funding Information:This research was supported in part by grants from the US-Israel Binational Science Foundation, and the KAMEA Fellowship program of the Ministry of Absorption of the State of Israel. D.b.-A. thanks the NSF (PHY-9820569) for support.

### Funding

This research was supported in part by grants from the US-Israel Binational Science Foundation, and the KAMEA Fellowship program of the Ministry of Absorption of the State of Israel. D.b.-A. thanks the NSF (PHY-9820569) for support.

Funders | Funder number |
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Ministry of Absorption of the State of Israel | |

US-Israel Binational Science Foundation | |

National Science Foundation | PHY-9820569 |

## Keywords

- Directed polymers
- Disordered systems
- Random walks