Phase diagram of the interacting partially directed self-avoiding walk attracted by a vertical wall

TL;DR

This paper rigorously analyzes the phase diagram of an interacting partially-directed self-avoiding walk attracted by a vertical wall, revealing a surface transition within the collapsed phase and providing precise asymptotics of the partition function.

Contribution

It proves the existence of a surface transition inside the collapsed phase and confirms physicists' conjecture about the polymer's partial attachment to the wall.

Findings

Identification of a surface transition within the collapsed phase.

Confirmation of physicists' conjecture on polymer-wall attachment.

Sharp asymptotics of the partition function in the collapsed phase.

Abstract

In the present paper, we consider the interacting partially-directed self-avoiding walk (IPDSAW) attracted by a vertical wall. The IPDSAW was introduced by Zwanzig and Lauritzen (J. Chem. Phys., 1968) as a manner of investigating the collapse transition of a homopolymer dipped in a repulsive solvent. We prove in particular that a surface transition occurs inside the collapsed phase between (i) a regime where the attractive vertical wall does not influence the geometry of the polymer and (ii) a regime where the polymer is partially attached at the wall on a length that is comparable to its horizontal extension, modifying its asymptotic Wulff shape. The latter rigorously confirms the conjecture exposed by physicists in (Physica A: Stat. Mech. \\& App., 2002). We push the analysis even further by providing sharp asymptotics of the partition function inside the collapsed phase.