A solvable non-directed model of polymer adsorption

TL;DR

This paper introduces a new solvable model of polymer adsorption using prudent walks on a square lattice, revealing a first-order transition at a critical surface interaction.

Contribution

It presents the first exact solution for a non-directed prudent walk model of polymer adsorption, expanding understanding beyond traditional directed models.

Findings

Exact generating functions and free energies derived

Identified a first-order adsorption transition

Characterized surface densities at criticality

Abstract

Prudent walks are self-avoiding walks which cannot step towards an already occupied vertex. We introduce a new model of adsorbing prudent walks on the square lattice, which start on an impenetrable surface and accrue a fugacity $a$ with each step along the surface. These are different to other exactly solved models of polymer adsorption, like Dyck paths, Motzkin paths and partially-directed walks, in that they are not trivially directed - they are able to step in all lattice directions. We calculate the generating functions, free energies and surface densities for this model and observe a first-order adsorption transition at the critical value of the surface interaction.