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

TL;DR

This paper analyzes a 1+1 dimensional self-interacting, partially directed self-avoiding walk model, precisely characterizing the free energy near the collapse transition and describing the geometric structure of the collapsed phase.

Contribution

It provides the asymptotic behavior of the free energy near criticality and describes the macroscopic geometry of the collapsed phase, including the Wulff shape.

Findings

Free energy near critical point scales as rac{/2}(eta_c-\u03b5) d gamma b5^{3/2}

Collapsed phase features a macroscopic bead formed by long vertical stretches

Rescaled envelope converges to a deterministic Wulff shape

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 $f(\beta_c-\epsilon)\sim \gamma \epsilon^{3/2}$ 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, we identify the horizontal extension of the random walk inside the collapsed phase and we establish the convergence of the rescaled envelope of the macroscopic bead towards a deterministic Wulff shape.