Dimensional reduction for directed branched polymers

Dimensional reduction occurs when the critical behaviour 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⊥(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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