Fields for Trees and Forests
Sergio Caracciolo


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.

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Elenco dei partecipanti al convegno di Bari.

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