Fermionic field theory for trees and forests

TL;DR

This paper generalizes Kirchhoff's matrix-tree theorem by representing combinatorial objects like unrooted spanning forests through non-Gaussian Grassmann integrals, linking them to well-studied models in statistical physics.

Contribution

It introduces a Grassmann integral representation for unrooted spanning forests and connects it to the N-vector and sigma models, revealing asymptotic freedom in 2D.

Findings

Representation of forests via Grassmann integrals

Mapping to N-vector and sigma models

Perturbative asymptotic freedom in 2D

Abstract

We prove a generalization of Kirchhoff's matrix-tree theorem in which a large class of combinatorial objects are represented by non-Gaussian Grassmann integrals. As a special case, we show that unrooted spanning forests, which arise as a q \to 0 limit of the Potts model, can be represented by a Grassmann theory involving a Gaussian term and a particular bilocal four-fermion term. We show that this latter model can be mapped, to all orders in perturbation theory, onto the N-vector model at N=-1 or, equivalently, onto the sigma-model taking values in the unit supersphere in R^{1|2}. It follows that, in two dimensions, this fermionic model is perturbatively asymptotically free.