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

Abstract

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 $L\in\mathbb{N}$ diverges, we prove that the set of occupied sites, rescaled horizontally by $L^{2/3}$ and vertically by $L^{1/3}$ 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 $a_{1}$ the time at which its geometric area reaches $1$. The modulus of $B$ up to $a_{1}$ gives 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).