Correlation functions and critical behaviour on fluctuating geometries

We study the two–point correlation function in the model of branched polymers and its relation to the critical behaviour of the model. We show that the correlation function has a universal scaling form in the generic phase with the only scale given by the size of the polymer. We show that the origin of the singularity of the free energy at the critical point is different from that in the standard statistical models. The transition is related to the change of the dimensionality of the system. The notion of correlation length is of a paramount importance in discrete field theoretical models. To define the continuum limit one requires correlation length to be infinite. This can be achieved by tuning the system to the continuous phase transition. Consider as an example a spin system. The correlation length can be defined by the behaviour of the connected correlation function :