Long linear polymer molecules in dilute solution are believed to undergo a collapse transition from an open coil to a compact ball when the temperature is lowered, or the solvent quality deteriorates. This phenomenon has been observed by light scatteriing and viscosity measurements though care must be taken to work at very low concentration and at very high molecular weight to observe the effect.
A natural model for this phenomenon is a self-avoiding walk on a lattice with an additional vertex-vertex interaction term, representing the monomer-monomer interaction in solution. If this interaction is sufficiently attractive then one expects collapse to occur. Although this model is easy to explain it is not easy to solve! At the present time there is no proof of the existence of the limiting free energy though this can be shown for the corresponding model of lattice polygons. In both cases there is no proof that the limiting free energy has a singularity. Most of the information we have about this model is from numerical methods such as Monte Carlo and exact enumeration coupled with series analysis.