A SHAPE THEOREM FOR THE SCALING LIMIT OF THE IPDSAW AT CRITICALITY

• 成果类型: Article
• 署名作者: Carmona, Philippe; Petrelis, Nicolas
• 署名单位: Nantes Universite
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/18-AAP1396
• 发表日期: 2019
• 页码: 875-930
• 关键词: transition; models

摘要

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 is an element of 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 Petrelis (2017).