The aim of this paper is twofold:

- To give an elementary and self-contained proof of an explicit formula for the free energy for a general class of polymer chains interacting with an environment through periodic potentials. This generalizes a result in [Bolthausen and Giacomin, AAP 2005] in which the formula is derived by using the Donsker-Varadhan Large Deviations theory for Markov chains. We exploit instead tools from renewal theory.

- To identify the infinite volume limits of the system. In particular, in the different regimes we encounter transient, null recurrent and positive recurrent processes (which correspond to delocalized, critical and localized behaviors of the trajectories). This is done by exploiting the sharp estimates on the partition function of the system obtained by the renewal theory approach.

The precise characterization of the infinite volume limits of the system exposes a non-uniqueness problem. We will however explain in detail how this (at first) surprising phenomenon is instead due to the presence of a first-order phase transition.

Caravenna, F; Giacomin, G; Zambotti, L (2007). *Infinite volume limits of polymer chains with periodic charges.* Markov Processes and Related Fields, 13(4):697-730.

## Abstract

The aim of this paper is twofold:

- To give an elementary and self-contained proof of an explicit formula for the free energy for a general class of polymer chains interacting with an environment through periodic potentials. This generalizes a result in [Bolthausen and Giacomin, AAP 2005] in which the formula is derived by using the Donsker-Varadhan Large Deviations theory for Markov chains. We exploit instead tools from renewal theory.

- To identify the infinite volume limits of the system. In particular, in the different regimes we encounter transient, null recurrent and positive recurrent processes (which correspond to delocalized, critical and localized behaviors of the trajectories). This is done by exploiting the sharp estimates on the partition function of the system obtained by the renewal theory approach.

The precise characterization of the infinite volume limits of the system exposes a non-uniqueness problem. We will however explain in detail how this (at first) surprising phenomenon is instead due to the presence of a first-order phase transition.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | 2007 |

Deposited On: | 07 Dec 2009 14:17 |

Last Modified: | 05 Apr 2016 13:23 |

Publisher: | Polomat |

ISSN: | 1024-2953 |

Official URL: | http://mech.math.msu.su/~malyshev/mprf.htm |

Related URLs: | http://arxiv.org/abs/math/0604426 http://www.ams.org/mathscinet-getitem?mr=2381597 |

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