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Dimensional Reduction for Directed Branched Polymers

John Z. Imbrie
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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: $γ(D+1)=α(D)$ (the exponent describing the singularity of the pressure), and $ν_{\perp}(D+1)=ν(D)$ (the correlation length exponent of the repulsive gas). It also leads to the relation $θ(D+1)=1+σ(D)$, where $σ(D)$ is the Yang-Lee edge exponent. We derive exact expressions for the number of DBP of size N in two dimensions.

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John Z. Imbrie. 2004-03-15. Dimensional Reduction for Directed Branched Polymers. https://doi.org/10.1088/0305-4470%2F37%2F12%2Fl03

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