Dimensional Reduction for Directed Branched Polymers
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 (D+1)=(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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