The Annals of Applied Probability

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

Philippe Carmona and Nicolas Pétrélis

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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).

Article information

Ann. Appl. Probab., Volume 29, Number 2 (2019), 875-930.

Received: September 2017
Revised: March 2018
First available in Project Euclid: 24 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B26: Phase transitions (general) 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Polymer collapse phase transition shape theorem Brownian motion Local limit theorem


Carmona, Philippe; Pétrélis, Nicolas. A shape theorem for the scaling limit of the IPDSAW at criticality. Ann. Appl. Probab. 29 (2019), no. 2, 875--930. doi:10.1214/18-AAP1396.

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