Dimensional reduction for directed branched polymers

Dimensional Reduction for Directed Branched Polymers

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)...

Branched Polymers and Dimensional Reduction

We establish an exact isomorphism between self-avoiding branched polymers in D + 2 continuum dimensions and the hard-core continuum gas at negative activity. We review conjectures and results on critical exponents for D + 2 = 2, 3, 4 and show that they are corollaries of our result. We explain the connection (first proposed by Parisi and Sourlas) between branched polymers in D + 2 dimensions an...

Lecture on Branched Polymers and Dimensional Reduction

This is a pedagogical account of the some of the recent results of Brydges and Imbrie, described from the point of view of Grassmann integration. Some simple extensions are pointed out. Historical Introduction The subject of dimensional reduction, in the context considered in this lecture, has a long and rather peculiar history. The strangest aspect of all is that the simplest formulation of th...

Flory approximation for directed branched polymers and directed percolation

Dimensional Reduction and Crossover to Mean-Field Behavior for Branched Polymers

This article will review recent results on dimensional reduction for branched polymers, and discuss implications for critical phenomena. Parisi and Sourlas argued in [PS81] that branched polymers fall into the universality class of the Yang-Lee edge in two fewer dimensions. Brydges and I have proven in [BI01] that the generating function for self-avoiding branched polymers in D + 2 continuum di...