Generating functions for trees, forests and unicyclics on finite geometries

A new method for proving Kirchhoff Matrix-Tree theorem, involving combinatorics of Grassmann variables, is developed. The theorem allows to evaluate the partition function of spanning trees on a given weighted graph, which emerges both in a limit of Potts Model, and of a free scalar fermion. The method generalizes to other counting problems beyond spanning trees: forests, hyper-trees and hyper-forests, and collections of unicyclic graphs, whose counterparts are more general cases of the related physical models. In particular, spanning forests correspond on Potts to consider the whole series in q, instead of a limit q = 0, while on the free-fermion theory correspond to a OSP(1|2) theory with variables on a supersphere of radius 1/q, instead of its linearized limit of infinite radius. Ward identities for OSP(1|2) symmetry correspond to combinatorial identities for connectivity patterns among vertices on the forest. For periodic graphs in 2 dimensions special features emerge. A Renormalization-group analysis suggests that the Spanning Tree theory is a fixed point, w.r.t. parameter q, marginally unstable in the q > 0 physical region, thus showing Asymptotic Freedom, the crucial ingredient of QCD confinement, in a specially simpler template model.