Interacting Partially Directed Self Avoiding Walk. From Phase Transition To The Geometry Of The Collapsed Phase

In this paper, we investigate a model for a 1 1 dimensional self interacting and partially directed self-avoiding walk, usually referred to by the acronym IPDSAW. The interaction intensity and the free energy of the system are denoted by beta and f, respectively. The IPDSAW is known to undergo a collapse transition at beta(c). We provide the precise asymptotic of the free energy close to criticality, that is, we show that f (beta(c) - epsilon) similar to gamma epsilon(3/2) where gamma is computed explicitly and interpreted in terms of an associated continuous model. We also establish some path properties of the random walk inside the collapsed phase (beta > beta(c)). We prove that the geometric conformation adopted by the polymer is made of a succession of long vertical stretches that attract each other to form a unique macroscopic bead and we establish the convergence of the region occupied by the path properly rescaled toward a deterministic Wulff shape.