  We analyze a generalization of the hard sphere dipole system in two
dimensions in which the interaction range of the interaction can be varied. We
focus on the system in the limit the interaction becomes increasingly
short-ranged, while the temperature becomes low. By using a cluster expansion
and taking advantage of low temperatures to perform saddle-point
approximations, we argue that a well defined double limit exists in which the
only structures which contribute to the free energy are chains. We then argue
that the dominance of chain structures is equivalent to the dominance of chain
diagrams in a cluster expansion, but only if the expansion is performed around
a hard sphere system (rather than the standard ideal gas). We show that this
leads to non-standard factorization rules for diagrams, and use this to
construct a closed-form expression for the free energy at low densities. We
then compare this construction to several models previously developed for the
hard sphere dipole system in the regime where chain structures dominate, and
argue that the comparison provides evidence in favor of one model over the
others. We also use this construction to incorporate some finite density
effects though the hard sphere radial distribution function, and analyze the
impact of these effects on chain length and the equation of state.
