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

In this paper we give a complete characterization of the scaling limit of the critical Interacting Partially Directed Self-Avoiding Walk (IPDSAW) introduced in Zwanzig and Lauritzen [J. Chem. Phys. 48 (1968) 3351].As the system size diverges, we prove that the set of occupied sites, rescaled horizontally by L2/3 and vertically by L1/3 converges in law for the Hausdorff distance toward a nontrivial random set.This limiting set is built with a Brownian motion B conditioned to come back at the origin at a1 the time at which its geometric area reaches 1.The modulus of B up to a1gives the height of the limiting set, while its center of mass process is an independent Brownian motion.Obtaining the shape theorem requires to derive a functional central limit theorem for the excursion of a random walk with Laplace symmetric increments conditioned on sweeping a prescribed geometric area.This result is proven in a companion paper Carmona and Pétrélis (2017).