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April 2019 A shape theorem for the scaling limit of the IPDSAW at criticality
Philippe Carmona, Nicolas Pétrélis
Ann. Appl. Probab. 29(2): 875-930 (April 2019). DOI: 10.1214/18-AAP1396

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).

Citation

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Philippe Carmona. Nicolas Pétrélis. "A shape theorem for the scaling limit of the IPDSAW at criticality." Ann. Appl. Probab. 29 (2) 875 - 930, April 2019. https://doi.org/10.1214/18-AAP1396

Information

Received: 1 September 2017; Revised: 1 March 2018; Published: April 2019
First available in Project Euclid: 24 January 2019

zbMATH: 07047441
MathSciNet: MR3910020
Digital Object Identifier: 10.1214/18-AAP1396

Subjects:
Primary: 60K35
Secondary: 82B26, 82B41

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 2 • April 2019
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