Interacting partially directed self avoiding walk. From phase transition to the geometry of the collapsed phase
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Philippe Carmona [1] ; Gia Bao Nguyen [1] ; Nicolas Pétrélis [2]
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[1] University of Nantes
University of Nantes
Arrondissement de Nantes, Francia
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[2] Universidad de Chile
Universidad de Chile
Santiago, Chile
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 5, 2016, págs. 3234-3290
- Idioma: inglés
- DOI: 10.1214/15-AOP1046
- Enlaces
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Resumen
In this paper, we investigate a model for a 1+11+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 ββ and ff, respectively. The IPDSAW is known to undergo a collapse transition at βcβc. We provide the precise asymptotic of the free energy close to criticality, that is, we show that f(βc−ε)∼γε3/2f(βc−ε)∼γε3/2 where γγ 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 (β>βc)(β>βc). 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.
