Polygons in restricted geometries subjected to infinite forces

TL;DR

This paper studies self-avoiding polygons in a restricted infinite tube subjected to large stretching or compressive forces, deriving their asymptotic free energy and exploring a conjecture relating to Hamiltonian polygons.

Contribution

It derives the asymptotic free energy of polygons under extreme forces and verifies a conjecture linking compressive force limits to Hamiltonian polygons for small tube sizes.

Findings

Asymptotic free energy forms for large positive and negative forces.

Verification of the conjecture for small tube sizes using transfer-matrix methods.

Identification of the relationship between force limits and Hamiltonian polygons.

Abstract

We consider self-avoiding polygons in a restricted geometry, namely an infinite $L\times M$ tube in $\mathbb Z^3$. These polygons are subjected to a force $f$, parallel to the infinite axis of the tube. When $f>0$ the force stretches the polygons, while when $f<0$ the force is compressive. We obtain and prove the asymptotic form of the free energy in both limits $f\to\pm\infty$. We conjecture that the $f\to-\infty$ asymptote is the same as the limiting free energy of "Hamiltonian" polygons, polygons which visit every vertex in a $L\times M\times N$ box. We investigate such polygons, and in particular use a transfer-matrix methodology to establish that the conjecture is true for some small tube sizes