Phase transition in fluctuating branched geometry

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We study grand--canonical and canonical properties of the model of branched polymers proposed in 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 0.3237525... is the first known example of positive γ 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 γ's in the range (0,1/2] and changes the order of the transition.