Randomly Branched Polymers and Conformal Invariance

TL;DR

This paper investigates the conformal invariance properties of randomly branched polymers in two dimensions, showing that the natural formulations are inconsistent and that numerical results contradict conformal invariance predictions.

Contribution

The paper demonstrates that the field theory for randomly branched polymers is not conformally invariant in two dimensions and highlights inconsistencies in natural conformal formulations.

Findings

Natural conformal formulations lead to inconsistencies.

The free field theory is not classically conformally invariant.

Numerical enumeration shows $ heta(eta)$ is not linear in $1/eta$.

Abstract

We argue that the field theory that descibes randomly branched polymers is not generally conformally invariant in two dimensions at its critical point. In particular, we show (i) that the most natural formulation of conformal invariance for randomly branched polymers leads to inconsistencies; (ii) that the free field theory obtained by setting the potential equal to zero in the branched polymer field theory is not even classically conformally invariant; and (iii) that numerical enumerations of the exponent $\theta (\alpha )$, defined by $T_N(\alpha )\sim \lambda^NN^{-\theta (\alpha ) +1}$, where $T_N(\alpha )$ is number of distinct configuratations of a branched polymer rooted near the apex of a cone with apex angel $\alpha$, indicate that $\theta (\alpha )$ is not linear in $1/\alpha$ contrary to what conformal invariance leads one to expect.