Surface transition in the collapsed phase of a self-interacting walk adsorbed along a hard wall

Abstract

The present paper is dedicated to the 2-dimensional Interacting Partially Directed Self Avoiding Walk constrained to remain in the upper-half plane and interacting with the horizontal axis. This model has originally been introduced to investigate the behavior of a homopolymer dipped in a poor solvent and adsorbed along a horizontal hard wall. It is known to undergo a collapse transition between an extended phase, inside which typical configurations of the polymer have a large horizontal extension (comparable to their total size), and a collapsed phase inside which the polymer looks like a globule. It is conjectured in the physics literature (see, e.g., (Phys. A, Stat. Mech. Appl. 318 (2002) 171) or (Phys. Rev. E 65 (2002) 056124)) that inside the collapsed phase, a surface transition occurs between an adsorbed-collapsed regime where the bottommost layer of the globule is pinned at the hard wall, and a desorbed-collapsed regime where the globule wanders away from the wall. In the present paper, we consider a simplified “single-bead” version of the model, for which we establish rigorously the existence of the surface transition and exhibit its associated critical curve. To that aim, we display some sharp asymptotics of the partition function of this simplified model within the collapsed phase.