Fields, trees, and forests

Abstract

A large class of combinatorial objects related to trees and forests can be represented by non- Gaussian Grassmann integrals, generalizing the Kirchhoff matrix-tree theorem. In particular, unrooted spanning forests with weight a per tree, which also arise as particular Q → 0 limits of the Potts model, can be represented by a Grassmann theory involving a Gaussian term and a particular bilocal four-fermion term. 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 hemi-supersphere in ℝ½. In two dimensions, the sigma model generically is either massive or flows to a critical symplectic fermion theory. However, the antiferromagnetic critical point of the Potts model corresponds to another critical point of the forest model (at negative a). The corresponding conformal theory on the square lattice (with a = −4) has a non-linearly realized OSP(2|2) = SL(1|2) symmetry, and involves non-compact degrees of freedom, with a continuous spectrum of critical exponents.