支化聚合物的降维与平均场行为的交叉
Dimensional Reduction and Crossover to Mean-Field Behavior for Branched Polymers
摘要：
本文将回顾支化聚合物降维的最新研究成果，并讨论其对临界现象的影响。Parisi和Sourlas在1981年提出，支化聚合物在两个更少的维度上属于杨-李边的普适类。Brydges和我在[math-ph/0107005]中证明，D+2连续维中自避免支化聚合物的母函数与D维中负活度时硬核连续气体的压力成正比（属于Yang-Lee或$i ϕ^3$类）。我将描述这种等价性是如何从支化聚合物模型的潜在超对称性中产生的。- 我还将使用降维来分析二维支化聚合物到其平均场极限的交叉，并表明缩放是由艾里函数给出的（与[cond-mat/0107223]中相同）。
英文摘要：
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 $i ϕ^3$ 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]).
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