Annals of Probability

Interacting partially directed self avoiding walk. From phase transition to the geometry of the collapsed phase

Philippe Carmona, Gia Bao Nguyen, and Nicolas Pétrélis

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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.

Article information

Ann. Probab., Volume 44, Number 5 (2016), 3234-3290.

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

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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 variational formula Wulff shape


Carmona, Philippe; Nguyen, Gia Bao; Pétrélis, Nicolas. Interacting partially directed self avoiding walk. From phase transition to the geometry of the collapsed phase. Ann. Probab. 44 (2016), no. 5, 3234--3290. doi:10.1214/15-AOP1046.

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