Exact Solution of the Discrete (1+1)-dimensional RSOS Model in a Slit with Field and Wall Interactions

TL;DR

This paper provides an exact solution to a linear RSOS model in a slit with field and wall interactions, revealing complex mathematical structures and generalizing previous models involving lattice paths and osmotic pressure.

Contribution

It extends the exact solution of the RSOS model to include a slit geometry with interactions, connecting to $q$-orthogonal polynomials and generalizing prior work on related lattice path models.

Findings

Solution involves basic hypergeometric functions ${}_3 ext{phi}_2$

Generalizes previous half-plane RSOS solutions to slit geometry

Links RSOS model to lattice polymer models with interactions

Abstract

We present the solution of a linear Restricted Solid--on--Solid (RSOS) model confined to a slit. We include a field-like energy, which equivalently weights the area under the interface, and also include independent interaction terms with both walls. This model can also be mapped to a lattice polymer model of Motzkin paths in a slit interacting with both walls and including an osmotic pressure. This work generalises previous work on the RSOS model in the half-plane which has a solution that was shown recently to exhibit a novel mathematical structure involving basic hypergeometric functions ${}_3\phi_2$. Because of the mathematical relationship between half-plane and slit this work hence effectively explores the underlying $q$-orthogonal polynomial structure to that solution. It also generalises two other recent works: one on Dyck paths weighted with an osmotic pressure in a slit and another concerning Motzkin paths without an osmotic pressure term in a slit.