Fermionic field theory for trees and forests
摘要
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.
引用
@article{arxiv.cond-mat/0403271,
title = {Fermionic field theory for trees and forests},
author = {Sergio Caracciolo and Jesper Lykke Jacobsen and Hubert Saleur and Alan D. Sokal and Andrea Sportiello},
journal= {arXiv preprint arXiv:cond-mat/0403271},
year = {2009}
}
备注
Revtex4, 4 pages. Version 2 (published in PRL) makes slight improvements in the exposition