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 $\phi^3$ theory with a single mass insertion, not those for the $\phi^3$ theory themselves. In particular, the two-point function behaves as $1/p^4$, not as $1/p^2$, in the scaling region. This behavior is consistent with the fact that the Hausdorff dimension of the branched polymer is four.