Caracciolo, S; Jacobsen, JL; Saleur, H; Sokal, AD; Sportiello, A; (2004) Fermionic field theory for trees and forests. PHYS REV LETT , 93 (8) , Article 080601. 10.1103/PhysRevLett.93.080601.

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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-->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-1parallel to2. It follows that, in two dimensions, this fermionic model is perturbatively asymptotically free.

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