2006/04/19 by Francesco Caravenna, Caravenna, Francesco, Giambattista Giacomin +3
Materials Science · Mathematics · Physics and Astronomy · #60K35 #82B41 #82B44 #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Material Dynamics and Properties #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR #msc:60K35 #msc:82B41 #msc:82B44
paper · pdf · doi:10.48550/arxiv.math/0604426
27 pages, 2 figures
arxiv created 2006/04/19 · openalex publication_date 2006/04/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.