Branched Polymers

TL;DR

This paper provides an elementary calculation of the volume of branched polymers in 2D and 3D, reveals new identities, and demonstrates that random 3D branched polymers have diameter proportional to the square root of their size.

Contribution

It introduces a simplified method for calculating the volume of branched polymers and uncovers new identities, enabling exact random sampling.

Findings

Volume formulas for branched polymers in 2D and 3D

New identities related to branched polymers

Random 3D branched polymers have diameter of order √n

Abstract

Building on and from the work of Brydges and Imbrie, we give an elementary calculation of the volume of the space of branched polymers of order $n$ in the plane and in 3-space. Our development reveals some more general identities, and allows exact random sampling. In particular we show that a random 3-dimensional branched polymer of order $n$ has diameter of order $\sqrt{n}$.