Polymer Statistics and Fermionic Vector Models

We consider a variation of O(N)-symmetric vector models in which the vector components are Grassmann numbers. We show that these theories generate the same sort of random polymer models as the O(N) vector models and that they lie in the same universality class in the large-N limit. We explicitly construct the double-scaling limit of the theory and show that the genus expansion is an alternating Borel summable series that otherwise coincides with the topological expansion of the bosonic models. We also show how the fermionic nature of these models leads to an explicit solution even at finite-N for the generating functions of the number of random polymer configurations. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. The statistical mechanics of randomly branching polymers (sometimes called discrete filamentary surfaces) is of interest in condensed matter physics where the connected polymer chains can be thought of as describing molecules. Random polymers have also been of interest as examples of random geometry systems which in many cases are exactly solvable and which have been argued to represent a certain dimensionally-reduced phase of Polyakov string theory in target space dimensions D > 1 [1, 2]. They therefore serve as toy models for more complicated higher dimensional problems such as statistical models of discretized surfaces which have been studied in the context of string theory and lower dimensional quantum gravity (see [3] for a review). O(N)-symmetric vector field theories [4] provide non-perturbative models of randomly branching chains where the order in the 1/N expansion coincides with the genus, or number of loops in the molecules of the ensemble. These models exhibit phase transitions in the large-N limit at which infinitely long polymers dominate the statistical sum and a continuum limit analogous to that found in the matrix model representation of random surface theories is reached [5]–[9]. The simplicity of the random polymers as compared to random surfaces allows a more explicit solution, even in dimensions D > 1, where some of the ideas about scaling and other critical behaviour can be tested. These simpler polymer structures are reflected in the linearity of the vector field theories in contrast to the non-linearity of matrix ones. The multicritical series generated by these models give generalized statistical systems which interpolate between the Cayley tree at one end and the ordinary random walk at the other [1]. Some supersymmetric generalizations of the O(N) vector model have been studied in [10]. In this Letter we shall show that a vector theory with purely fermionic degrees of freedom can also be used to represent random polymers. This model exhibits the same critical behaviour in the large-N limit as the O(N) vector model, but it has a structure which is in some respects simpler. In particular, we will show that its genus expansion is an alternating series which is Borel summable and that it represents a rare example of a random geometry theory whose explicit solution can be written down even at finite N . It therefore provides an explicit, well-defined generating function for a given number of polymer configurations in a random polymer system. It can also be combined with other polymer models to generate new types of generating functions for random surface theories. For example, it can be combined with the O(N) vector model to study random polymer theories with either only even or only odd genera. Let us start by reviewing some features of the O(N) vector model with partition function [6, 7, 9] ZS(t, g;N) = ∫ N