Dimensional Reduction for Directed Branched Polymers

Details Dimensional Reduction for Directed Branched Polymers Dimensional Reduction for Directed Branched Polymers

2004-02-27

arxiv.org/abs/math-ph/0402074v2

J. Phys. A: Math. Gen. 37, L137--L142 (2004)

Physics

Mathematical Physics

7 pages, 1 eps figure, ref 24 corrected

Scientific paper

10.1088/0305-4470/37/12/L03

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.

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