- 03-244 John Z. Imbrie
- Dimensional Reduction and Crossover to Mean-Field Behavior for Branched Polymers
May 28, 03
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Abstract. 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 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 [C01]).