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 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 1n, n = 2, 3, …. We show that a slight modification of the model produces a continuos spectrum of γʹs in the range (0, 12] and changes the order of the transition.