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 → 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 σ-model taking values in the unit supersphere in R 1|2 . It follows that, in two dimensions, this fermionic model is perturbatively asymptotically free.PACS numbers: 05.50.+q, 02.10. Ox, 11.10.Hi, 11.10.Kk Kirchhoff's matrix-tree theorem [1] and its generalizations [2], which express the generating polynomials of spanning trees and rooted spanning forests in a graph as determinants associated to the graph's Laplacian matrix, play a central role in electrical circuit theory [3] and in certain exactly-soluble models in statistical mechanics [4,5]. Like all determinants, those arising in Kirchhoff's theorem can of course be rewritten as Gaussian integrals over fermionic (Grassmann) variables.In this Letter we prove a generalization of Kirchhoff's theorem in which a large class of combinatorial objects are represented by suitable non-Gaussian Grassmann integrals. Although these integrals can no longer be calculated in closed form, our identities allow the use of field-theoretic methods to shed new light on the critical behavior of the underlying geometrical models.As a special case, we show that unrooted spanning forests, which arise as a q → 0 limit of the q-state Potts model [6], can be represented by a Grassmann theory involving a Gaussian term and a particular bilocal four-fermion term. Furthermore, this latter model can be mapped, to all orders in perturbation theory,It follows that, in two dimensions, this fermionic model is perturbatively asymptotically free, in close analogy to (large classes of) two-dimensional σ-models and four-dimensional nonabelian gauge theories. Indeed, this fermionic model may, because of its great simplicity, be the most viable candidate for a rigorous nonperturbative proof of asymptotic freedom -a goal that has heretofore remained elusive in both σ-models and gauge theories.The plan of this Letter is as follows: First we prove some combinatorial identities involving Grassmann integrals, culminating in our general formula (12), and show how a special case yields unrooted spanning forests. Next we show that this latter model can be mapped onto the N -vector model at N = −1, and use this fact to deduce its renormalization-group (RG) flow at weak coupling. Finally, we conjecture the nonperturbative phase diagram in this model.Combinatorial Identities. Let G = (V, E) be a finite undirected graph with vertex set V and edge set E. Associate to each edge e a weight w e , which can be a real or complex number or, more generally, a formal algebraic variable. For i = j, let...

We revisit the problem of determining the Arctic curve in the six-vertex model with domain wall boundary conditions. We describe an alternative method, by which we recover the previously conjectured analytic expression in the square domain. We adapt the method to work for a large class of domains, and for other models exhibiting limit shape phenomena. We study in detail some examples, and derive, in particular, the Arctic curve of the six-vertex model in a triangoloid domain at the ice-point.