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

TL;DR

This paper reviews recent advances in dimensional reduction for branched polymers, highlighting their connection to the Yang-Lee edge universality class and analyzing the crossover to mean-field behavior using supersymmetry and Airy functions.

Contribution

It demonstrates the equivalence between branched polymers and the Yang-Lee edge in lower dimensions via supersymmetry and explores the crossover to mean-field behavior with explicit scaling functions.

Findings

Branched polymers in D+2 dimensions relate to the Yang-Lee edge in D dimensions.

Supersymmetry underpins the dimensional reduction equivalence.

Crossover scaling is described by an Airy function.

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 $i \phi^3$ 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 [cond-mat/0107223]).