Two-stage collapse of a polymer chain in two dimensions

I show that for a certain range in its parameter space, a self-attracting self-avoiding loop (SASAL) can be mapped onto the hull of a percolation cluster in two dimensions. The inside (outside) of this loop attracts itself with interaction energy ${\mathrm{\varepsilon }}_{+}$ (${\mathrm{\varepsilon }}_{\mathrm{-}}$). This mapping is used to argue that if ${\mathrm{\varepsilon }}_{+}$>${\mathrm{\varepsilon }}_{\mathrm{-}}$, the collapse of the SASAL occurs in not one but two stages as the temperature is reduced: the SASAL first collapses to form a ‘‘branched polymer’’ and then at lower temperatures collapses still further to become a compact globule.