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

TL;DR

This paper rigorously analyzes a 2D self-interacting walk model to identify a surface transition within the collapsed phase, distinguishing between adsorbed and desorbed globule regimes along a hard wall.

Contribution

It establishes the existence of a surface transition in the collapsed phase of a self-interacting walk model, providing sharp asymptotics for the partition function.

Findings

Identification of a surface transition between adsorbed and desorbed globules

Derivation of sharp asymptotics for the partition function

Rigorous proof of the transition's critical curve

Abstract

The present paper is dedicated to the 2-dimensional Interacting Partially Directed Self Avoiding Walk constrained to remain in the upper-half plan and interacting with the horizontal axis. The model has been introduced in \cite{F90} 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. In the present paper, we establish rigorously 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. To prove the existence of this surface transition and exhibit its associated critical curve, we display some sharp asymptotics of the partition function for a slightly simplified version of the model.