Fermionic field theory for trees and forests

Details Fermionic field theory for trees and forests Fermionic field theory for trees and forests

2004-03-10

arxiv.org/abs/cond-mat/0403271v2

Phys.Rev.Lett. 93 (2004) 080601

Physics

Condensed Matter

Statistical Mechanics

Revtex4, 4 pages. Version 2 (published in PRL) makes slight improvements in the exposition

Scientific paper

10.1103/PhysRevLett.93.080601

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.

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