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 dimensions and repulsive gases at negative activity in dimensions. This implies relations between exponents of the two models:
(the exponent describing the singularity of the pressure), and
(the correlation length exponent of the repulsive gas). It also leads to the relation
, where is the Yang-Lee edge exponent. We derive exact expressions for the number of DBP of size in two dimensions.

John Imbrie
2003-09-10