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In this paper, we investigate a model for a $1+1$ dimensionalself-interacting and partially directed self-avoiding walk, usually referred toby the acronym IPDSAW. The interaction intensity and the free energy of thesystem are denoted by $\beta$ and $f$, respectively. The IPDSAW is known toundergo a collapse transition at $\beta_c$. We provide the precise asymptoticof the free energy close to criticality, that is we show that$f(\beta_c-\epsilon)\sim \gamma \epsilon^{3/2}$ where $\gamma$ is computedexplicitly and interpreted in terms of an associated continuous model. We alsoestablish some path properties of the random walk inside the collapsed phase$(\beta>\beta_c)$. We prove that the geometric conformation adopted by thepolymer is made of a succession of long vertical stretches that attract eachother to form a unique macroscopic bead, we identify the horizontal extensionof the random walk inside the collapsed phase and we establish the convergenceof the rescaled envelope of the macroscopic bead towards a deterministic Wulffshape.

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