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In this paper we give a complete characterization of the scaling limit of thecritical Interacting Partially Directed Self-Avoiding Walk (IPDSAW) introducedin Zwanzig and Lauritzen (1968). As the system size $L$ diverges, we prove thatthe set of occupied sites, rescaled horizontally by $L^{2/3}$ and vertically by$L^{1/3}$ converges in law for the Hausdorff distance towards a non trivialrandom set. This limiting set is built with a Brownian motion $B$ conditionedto come back at the origin at $a_1$ the time at which its geometric areareaches $1$. The modulus of $B$ up to $a_1$ gives the height of the limitingset, while its center of mass process is an independent Brownian motion.  Obtaining the shape theorem requires to derive a functional central limittheorem for the excursion of a random walk with Laplace symmetric incrementsconditioned on sweeping a prescribed geometric area. This result is proven in acompanion paper arXiv:1709.06448.

A shape theorem for the scaling limit of the IPDSAW at criticality