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Interacting partially directed self-avoiding walk: a probabilistic perspective
Univ Nantes, Lab Math Jean Leray, UMR 6629, 2 Rue Houssiniere,BP 92208, F-44322 Nantes 03, France..
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Univ Nantes, Lab Math Jean Leray, UMR 6629, 2 Rue Houssiniere,BP 92208, F-44322 Nantes 03, France..
CNRS, Campus Jussieu,4 Pl Jussieu, F-75252 Paris 5, France.;Univ Paris 06, Lab Probabil & Modeles Aleatoires, Campus Jussieu,4 Pl Jussieu, F-75252 Paris 5, France..
2018 (English)In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 51, no 15, article id 153001Article, review/survey (Refereed) Published
Abstract [en]

We review some recent results obtained in the framework of the 2D interacting self-avoiding walk (ISAW). After a brief presentation of the rigorous results that have been obtained so far for ISAW we focus on the interacting partially directed self-avoiding walk (IPDSAW), a model introduced in Zwanzig and Lauritzen (1968 J. Chem. Phys. 48 3351) to decrease the mathematical complexity of ISAW. In the first part of the paper, we discuss how a new probabilistic approach based on a random walk representation (see Nguyen and Petrelis (2013 J. Stat. Phys. 151 1099-120)) allowed for a sharp determination of the asymptotics of the free energy close to criticality (see Carmona et al (2016 Ann. Probab. 44 3234-90)). Some scaling limits of IPDSAW were conjectured in the physics literature (see e.g. Brak et al (1993 Phys. Rev. E 48 2386-96)). We discuss here the fact that all limits are now proven rigorously, i.e. for the extended regime in Carmona and Petrelis (2016 Electron. J. Probab. 21 1-52), for the collapsed regime in Carmona et al (2016 Ann. Probab. 44 3234-90) and at criticality in Carmona and Petrelis (2017b arxiv:1709.06448). The second part of the paper starts with the description of four open questions related to physically relevant extensions of IPDSAW. Among such extensions is the interacting prudent self-avoiding walk (IPSAW) whose configurations are those of the 2D prudent walk. We discuss the main results obtained in Petrelis and Torri (2016 Ann. Inst. Henri Poincare D) about IPSAW and in particular the fact that its collapse transition is proven to exist rigorously.

Place, publisher, year, edition, pages
IOP PUBLISHING LTD , 2018. Vol. 51, no 15, article id 153001
Keywords [en]
polymer collapse, phase transition, Wulff shape, local limit theorem, scaling limit
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-225697DOI: 10.1088/1751-8121/aab15eISI: 000427720400001OAI: oai:DiVA.org:kth-225697DiVA, id: diva2:1196824
Note

QC 20180411

Available from: 2018-04-11 Created: 2018-04-11 Last updated: 2018-04-11Bibliographically approved

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