Phase transition in fluctuating branched geometry

Abstract

We study grand--canonical and canonical properties of the model of branched polymers proposed in \cite{adfo}. We show that the model has a fourth order phase transition and calculate critical exponents. At the transition the exponent $γ$ of the grand-canonical ensemble, analogous to the string susceptibility exponent of surface models, $γ\sim 0.3237525...$ is the first known example of positive $γ$ which is not of the form $1/n,\, n=2,3,\ldots$. We show that a slight modification of the model produces a continuos spectrum of $γ$'s in the range $(0,1/2]$ and changes the order of the transition.