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