Localization and perron-frobenius theory for directed polymers

Yuri Bakhtin, Konstantin Khanin

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)667-686
Number of pages20
JournalMoscow Mathematical Journal
Issue number4
StatePublished - 2010


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

ASJC Scopus subject areas

  • General Mathematics


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