Droplet models, renormalization group, and essential singularities at first-order phase transitions

We propose (in $d=2\mathrm{}$) a modified form of the droplet model studied by Langer and Fisher. We solve this model exactly and with the renormalization group. Both methods obtain an essential singularity of the form proposed by Klein, Wallace, and Zia; however, the renormalization group misses an additional singularity identical to that of the standard droplet model. We discuss the implications of this result for Ising models in $d=2\mathrm{}$. Higher dimensions are also discussed.