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

TL;DR

This paper characterizes the scaling limit of the critical Interacting Partially Directed Self-Avoiding Walk (IPDSAW), showing convergence to a random set built from a conditioned Brownian motion, revealing detailed geometric properties at criticality.

Contribution

It provides a complete description of the scaling limit of the critical IPDSAW, including the convergence of rescaled occupied sites to a Brownian motion conditioned on geometric area.

Findings

Rescaled occupied sites converge to a Brownian motion conditioned on area.

The limiting set's shape is determined by the modulus and center of mass of the conditioned Brownian motion.

A functional central limit theorem for a conditioned random walk is established in a companion paper.

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 (1968). As the system size $L$ 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 towards a non trivial 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 arXiv:1709.06448.