2009/09/12 by Yuri Bakhtin, Bakhtin, Yuri, Konstantin Khanin +1
Materials Science · Mathematics · Physics and Astronomy · #60K37 #82D30 #FOS: Mathematics #FOS: Physical sciences #Lanthanide and Transition Metal Complexes #Magnetism in coordination complexes #Mathematical Physics (math-ph) #Probability (math.PR) #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.PR #msc:60K37 #msc:82D30
paper · pdf · doi:10.48550/arxiv.0909.2293
25 pages
arxiv created 2009/09/12 · openalex publication_date 2009/09/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.