# Fermionic field theory for trees and forests

### Sergio Caracciolo 1, Jesper-Lykke Jacobsen 2, Hubert Saleur 3, 4, Alan D. Sokal 5, Andrea Sportiello 1

#### Physical Review Letters 93 (2004) 080601

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.

• 1. Dipartimento di Fisica (Milano),
Università degli studi di Milano
• 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI - Paris Sud
• 3. Service de Physique Théorique (SPhT),
CNRS : URA2306 – CEA : DSM/SPHT
• 4. Department of Physics and Astronomy,
University of Southern California
• 5. Department of Physics,
New York University