Scaling behaviors of branched polymers

We study the thermodynamic behavior of branched polymers. We first study random walks in order to clarify the thermodynamic relation between the canonical ensemble and the grand canonical ensemble. We then show that correlation functions for branched polymers are given by those for φ³ theory with a single mass insertion, not those for the φ³ theory themselves. In particular, the two-point function behaves as 1/p⁴, not as 1/p², in the scaling region. This behavior is consistent with the fact that the Hausdorff dimension of the branched polymer is 4. In the appendixes we derive the exact two-point correlation function at finite (but large) system size N, which is consistent with the analysis in the body of the paper.