Thin animals

Lattice animals provide a discretized model for the ?-transition displayed by branched polymers in solvent. Exact graph enumeration studies have given some indications that the phase diagram of such lattice animals may contain two collapsed phases as well as an extended phase. This has not been confirmed by studies using other means. We use the exact correspondence between the q ? 1 limit of an extended Potts model and lattice animals to investigate the phase diagram of lattice animals on ?³ random graphs of arbitrary topology ('thin' random graphs). We find that only a two-phase structure exists - there is no sign of a second collapsed phase. The random graph model is solved in the thermodynamic limit by saddle-point methods. We observe that the ratio of these saddle-point equations give precisely the fixed points of the recursion relations that appear in the solution of the model on the Bethe lattice by Henkel and Seno (1996 Phys. Rev. E 53 3662). This explains the equality of non-universal quantities such as the critical lines for the Bethe lattice and random graph ensembles.