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 $f(\beta_c-\epsilon)\sim \gamma \epsilon^{3/2}$ 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, 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.