### Abstract

We consider directed polymers in a random potential given by a deterministic profile with a strong maximum at the origin taken with random sign at each integer time.We study two main objects based on paths in this random potential. First, we use the random potential and averaging over paths to define a parabolic model via a random Feynman-Kac evolution operator. We show that for the resulting cocycle, there is a unique positive cocycle eigenfunction serving as a forward and pullback attractor. Secondly, we use the potential to define a Gibbs specification on paths for any bounded time interval in the usual way and study the thermodynamic limit and existence and uniqueness of an infinite volume Gibbs measure. Both main results claim that the local structure of interaction leads to a unique macroscopic object for almost every realization of the random potential.

Original language | English (US) |
---|---|

Pages (from-to) | 667-686 |

Number of pages | 20 |

Journal | Moscow Mathematical Journal |

Volume | 10 |

Issue number | 4 |

State | Published - 2010 |

### Fingerprint

### Keywords

- Directed polymers
- Localization
- Parabolic model
- Perron-frobenius theorem

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Moscow Mathematical Journal*,

*10*(4), 667-686.

**Localization and perron-frobenius theory for directed polymers.** / Bakhtin, Yuri; Khanin, Konstantin.

Research output: Contribution to journal › Article

*Moscow Mathematical Journal*, vol. 10, no. 4, pp. 667-686.

}

TY - JOUR

T1 - Localization and perron-frobenius theory for directed polymers

AU - Bakhtin, Yuri

AU - Khanin, Konstantin

PY - 2010

Y1 - 2010

N2 - We consider directed polymers in a random potential given by a deterministic profile with a strong maximum at the origin taken with random sign at each integer time.We study two main objects based on paths in this random potential. First, we use the random potential and averaging over paths to define a parabolic model via a random Feynman-Kac evolution operator. We show that for the resulting cocycle, there is a unique positive cocycle eigenfunction serving as a forward and pullback attractor. Secondly, we use the potential to define a Gibbs specification on paths for any bounded time interval in the usual way and study the thermodynamic limit and existence and uniqueness of an infinite volume Gibbs measure. Both main results claim that the local structure of interaction leads to a unique macroscopic object for almost every realization of the random potential.

AB - We consider directed polymers in a random potential given by a deterministic profile with a strong maximum at the origin taken with random sign at each integer time.We study two main objects based on paths in this random potential. First, we use the random potential and averaging over paths to define a parabolic model via a random Feynman-Kac evolution operator. We show that for the resulting cocycle, there is a unique positive cocycle eigenfunction serving as a forward and pullback attractor. Secondly, we use the potential to define a Gibbs specification on paths for any bounded time interval in the usual way and study the thermodynamic limit and existence and uniqueness of an infinite volume Gibbs measure. Both main results claim that the local structure of interaction leads to a unique macroscopic object for almost every realization of the random potential.

KW - Directed polymers

KW - Localization

KW - Parabolic model

KW - Perron-frobenius theorem

UR - http://www.scopus.com/inward/record.url?scp=79960987687&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79960987687&partnerID=8YFLogxK

M3 - Article

VL - 10

SP - 667

EP - 686

JO - Moscow Mathematical Journal

JF - Moscow Mathematical Journal

SN - 1609-3321

IS - 4

ER -