Polygons in restricted geometries subjected to infinite forces

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