Dimensional Reduction Formulas for Branched Polymer Correlation Functions

TL;DR

This paper reviews and generalizes the connection between branched polymer generating functions and hard-core gas correlation functions, deriving new identities and scaling limits relevant for understanding critical behavior in statistical physics.

Contribution

It extends previous results by deriving Ward identities, reduction formulas for multi-species systems, and the scaling form of the 2-point function in three dimensions.

Findings

Generating functions are proportional to correlation functions of the hard-core gas.

Derived Ward identities for branched polymer correlations.

Obtained the scaling form of the 2-point function in 3D.

Abstract

In [math-ph/0107005] we have proven 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. This result explains why the critical behavior of branched polymers should be the same as that of the $i \phi^3$ (or Yang-Lee edge) field theory in two fewer dimensions (as proposed by Parisi and Sourlas in 1981). - In this article we review and generalize the results of [math-ph/0107005]. We show that the generating functions for several branched polymers are proportional to correlation functions of the hard-core gas. We derive Ward identities for certain branched polymer correlations. We give reduction formulae for multi-species branched polymers and the corresponding repulsive gases. Finally, we derive the massive scaling limit for the 2-point function of the one-dimensional hard-core gas, and thereby obtain the scaling form of the 2-point function for branched polymers in three dimensions.