Phase transition in fluctuating branched geometry

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