Interacting partially directed self avoiding walk. From phase transition to the geometry of the collapsed phase

Abstract

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 β and f , respectively. The IPDSAW is known to undergo a collapse transition at βc. We provide the precise asymptotic of the free energy close to criticality, that is we show that f (βc − e) ∼ γe 3/2 where γ 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 (β > βc). 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, we identify the horizontal extension of the random walk inside the collapsed phase and we establish the convergence of the rescaled envelope of the macroscopic bead towards a deterministic Wulff shape.