Fermionic field theory for trees and forests

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.

Revtex4, 4 pages. Version 2 (published in PRL) makes slight improvements in the exposition

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Cited by in corpus (48)

• Logarithmic Conformal Field Theory: a Lattice Approach
• Anomalous dimensions in CFT with weakly broken higher spin symmetry
• Spanning forests and the q-state Potts model in the limit q \to 0
• The ABC (in any D) of Logarithmic CFT
• Indecomposability parameters in chiral Logarithmic Conformal Field Theory
• The arboreal gas and the supersphere sigma model
• Ferromagnetic phase transition for the spanning-forest model (q \to 0 limit of the Potts model) in three or more dimensions
• Six dimensional QCD at two loops
• Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians
• Operator content of the critical Potts model in d dimensions and logarithmic correlations
• Grassmann Integral Representation for Spanning Hyperforests
• LERW as an example of off-critical SLEs
• Global symmetry and conformal bootstrap in the two-dimensional $O(n)$ model
• Corner free energies and boundary effects for Ising, Potts and fully-packed loop models on the square and triangular lattices
• Geometric Algebra and Algebraic Geometry of Loop and Potts Models
• Explicit characterization of the identity configuration in an Abelian Sandpile Model
• Multiple and inverse topplings in the Abelian Sandpile Model
• Exact solution of the $2d$ dimer model: Corner free energy, correlation functions and combinatorics
• Non-scalar operators for the Potts model in arbitrary dimension
• Ginzburg-Landau Description and Emergent Supersymmetry of the $(3,8)$ Minimal Model
• Two-point connectivity of two-dimensional critical $Q-$ Potts random clusters on the torus
• Renormalization flow for unrooted forests on a triangular lattice
• Q-colourings of the triangular lattice: Exact exponents and conformal field theory
• Conformally invariant boundary conditions in the antiferromagnetic Potts model and the $SL(2,\mathbb{R})/U(1)$ sigma model
• Phase transition in the spanning-hyperforest model on complete hypergraphs
• The antiferromagnetic Potts model
• Phase diagram of the triangular-lattice Potts antiferromagnet
• Critical Field Theories with OSp$(1|2M)$ Symmetry
• Spanning Forests on Random Planar Lattices
• A randomized polynomial-time algorithm for the Spanning Hypertree Problem on 3-uniform hypergraphs
• Hyperforests on the Complete Hypergraph by Grassmann Integral Representation
• Spontaneous symmetry breaking in 2D supersphere sigma models and applications to intersecting loop soups
• Negatively correlated random variables and Mason's conjecture
• Crossover exponents, fractal dimensions and logarithms in Landau-Potts field theories
• Critical Behaviour of Spanning Forests on Random Planar Graphs
• A Flow in the Forest
• Bootstrap approach to geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: A study of the $s$-channel spectra
• Sphere free energy of scalar field theories with cubic interactions
• Dimer and fermionic formulations of a class of colouring problems
• Uniqueness of the infinite tree in low-dimensional random forests
• Higher Order and boundary Scaling Fields in the Abelian Sandpile Model
• Anomalous dimensions from conformal field theory: Generalized ${\varphi }^{2n+1}$ theories
• Some Aspects of c=-2 Theory
• Percolation transition for random forests in $d\ge 3$
• Dualities between fermionic theories and the Potts model
• Conformal field theory for annulus SLE: partition functions and martingale-observables
• An equivalence between gauge-twisted and topologically conditioned scalar Gaussian free fields
• Anomalous criticality coexists with giant cluster in the uniform forest model