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.

Fermionic field theory for trees and forests / S. Caracciolo, J. L. Jacobsen, H. Saleur, A. D. Sokal, A. Sportiello. - In: PHYSICAL REVIEW LETTERS. - ISSN 0031-9007. - 93:8(2004), pp. 080601/1-080601/4.

Fermionic field theory for trees and forests

S. Caracciolo
Primo
;
A. Sportiello
Ultimo
2004

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 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.
lattice field theory; lattice theory; quantum field theory; trees (mathematics)
Settore FIS/02 - Fisica Teorica, Modelli e Metodi Matematici
2004
Article (author)
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/16730
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