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

Abstract

This article will review recent results on dimensional reduction for branched polymers, and discuss implications for critical phenomena. Parisi and Sourlas argued in 1981 that branched polymers fall into the universality class of the Yang-Lee edge in two fewer dimensions. Brydges and I have proven in [math-ph/0107005] that the generating function for self-avoiding branched polymers in D+2 continuum dimensions is proportional to the pressure of the hard-core continuum gas at negative activity in D dimensions (which is in the Yang-Lee or $i{\varphi }^3$ class). I will describe how this equivalence arises from an underlying supersymmetry of the branched polymer model. - I will also use dimensional reduction to analyze the crossover of two-dimensional branched polymers to their mean-field limit, and to show that the scaling is given by an Airy function (the same as in [cond-mat/0107223]).

15 pages, 1 eps figure

References in corpus (4)

• Two Dimensional Gauge Theories Revisited
• Scale Invariance in disordered systems: the example of the Random Field Ising Model
• Identity of the universal repulsive-core singularity with Yang-Lee edge criticality
• Dimensional Reduction Formulas for Branched Polymer Correlation Functions

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• Fermionic field theory for trees and forests
• Evidence for Supersymmetry in the Random-Field Ising Model at D = 5
• Functional integral representations for self-avoiding walk
• Counting Lattice Animals in High Dimensions
• Series Expansion of the Percolation Threshold on Hypercubic Lattices
• Lecture on Branched Polymers and Dimensional Reduction
• Fisher Renormalization for Logarithmic Corrections
• Statistics of randomly branched polymers in a semi-space
• Scaling behaviour of lattice animals at the upper critical dimension