Dimensional Reduction for Directed Branched Polymers

TL;DR

This paper establishes an exact relationship between directed branched polymers in D+1 dimensions and repulsive gases in D dimensions, revealing a dimensional reduction in their critical behaviors and exponents.

Contribution

It provides the first exact mapping between directed branched polymers and repulsive gases, deriving relations between their critical exponents and explicit formulas in two dimensions.

Findings

Relations between critical exponents of DBP and gases in different dimensions

Exact expressions for the number of DBP of size N in two dimensions

Dimensional reduction links the behavior of complex polymers to simpler gas models

Abstract

Dimensional reduction occurs when the critical behavior of one system can be related to that of another system in a lower dimension. We show that this occurs for directed branched polymers (DBP) by giving an exact relationship between DBP models in D+1 dimensions and repulsive gases at negative activity in D dimensions. This implies relations between exponents of the two models: $\gamma(D+1)=\alpha(D)$ (the exponent describing the singularity of the pressure), and $\nu_{\perp}(D+1)=\nu(D)$ (the correlation length exponent of the repulsive gas). It also leads to the relation $\theta(D+1)=1+\sigma(D)$, where $\sigma(D)$ is the Yang-Lee edge exponent. We derive exact expressions for the number of DBP of size N in two dimensions.