Polymer Statistics and Fermionic Vector Models

TL;DR

This paper explores fermionic vector models with Grassmann components, demonstrating their equivalence to bosonic models in universality class, and provides explicit solutions for generating functions of polymer configurations.

Contribution

It introduces a fermionic variant of $O(N)$ vector models, showing their universality class and providing explicit finite-$N$ solutions for polymer configuration generating functions.

Findings

Fermionic models generate similar polymer models as bosonic ones.

The double-scaling limit reveals an alternating Borel summable genus expansion.

Explicit finite-$N$ solutions for polymer configuration generating functions are obtained.

Abstract

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.