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September 2016 Interacting partially directed self avoiding walk. From phase transition to the geometry of the collapsed phase
Philippe Carmona, Gia Bao Nguyen, Nicolas Pétrélis
Ann. Probab. 44(5): 3234-3290 (September 2016). DOI: 10.1214/15-AOP1046

Abstract

In this paper, we investigate a model for a $1+1$ dimensional self-interacting and partially directed self-avoiding walk, usually referred to by the acronym IPDSAW. The interaction intensity and the free energy of the system are denoted by $\beta$ and $f$, respectively. The IPDSAW is known to undergo a collapse transition at $\beta_{c}$. We provide the precise asymptotic of the free energy close to criticality, that is, we show that $f(\beta_{c}-\varepsilon)\sim\gamma\varepsilon^{3/2}$ where $\gamma$ is computed explicitly and interpreted in terms of an associated continuous model. We also establish some path properties of the random walk inside the collapsed phase $(\beta>\beta_{c})$. We prove that the geometric conformation adopted by the polymer is made of a succession of long vertical stretches that attract each other to form a unique macroscopic bead and we establish the convergence of the region occupied by the path properly rescaled toward a deterministic Wulff shape.

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Philippe Carmona. Gia Bao Nguyen. Nicolas Pétrélis. "Interacting partially directed self avoiding walk. From phase transition to the geometry of the collapsed phase." Ann. Probab. 44 (5) 3234 - 3290, September 2016. https://doi.org/10.1214/15-AOP1046

Information

Received: 1 February 2014; Revised: 1 July 2015; Published: September 2016
First available in Project Euclid: 21 September 2016

zbMATH: 1360.60173
MathSciNet: MR3551196
Digital Object Identifier: 10.1214/15-AOP1046

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

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 5 • September 2016
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