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 $\beta$ 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 $f(\beta_{c}-\varepsilon)\sim\gamma\varepsilon^{3/2}$ 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.