Thin Animals

TL;DR

This paper investigates the phase diagram of lattice animals using an extended Potts model on thin random graphs, finding only a two-phase structure and confirming the absence of a second collapsed phase.

Contribution

It demonstrates that the phase diagram of lattice animals on thin random graphs contains only two phases, resolving previous ambiguities about a potential second collapsed phase.

Findings

Only two phases are observed in the model.

The saddle point equations match fixed points of Bethe lattice recursion.

Critical lines are consistent between Bethe lattice and random graph models.

Abstract

Lattice animals provide a discretized model for the theta 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 phi-cubed 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. This explains the equality of non-universal quantities such as the critical lines for the Bethe lattice and random graph ensembles.