Scaling Behaviors of Branched Polymers

TL;DR

This paper investigates the thermodynamic properties of branched polymers, revealing their correlation functions behave as in a specific field theory and confirming their Hausdorff dimension is four.

Contribution

It establishes the relation between branched polymers and $^3$ theory, showing the two-point function scales as $1/p^4$, indicating a Hausdorff dimension of four.

Findings

Correlation functions match $^3$ theory with a mass insertion

Two-point function scales as $1/p^4$ in the scaling region

Hausdorff dimension of branched polymers is four

Abstract

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.