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 inD +2 continuum dimensions is proportional to the pressure of the hard-core continuum gas at negative activity inDdimensions (which is in the Yang-Lee ori \(\phi ^3 \) 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[Car01]).
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Imbrie, J.Z. (2003). Dimensional Reduction and Crossover to Mean-Field Behavior for Branched Polymers. In: Iagolnitzer, D., Rivasseau, V., Zinn-Justin, J. (eds) International Conference on Theoretical Physics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7907-1_35
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