Grassmann Integral Representation for Spanning Hyperforests

Abstract

Given a hypergraph G, we introduce a Grassmann algebra over the vertex set, and show that a class of Grassmann integrals permits an expansion in terms of spanning hyperforests. Special cases provide the generating functions for rooted and unrooted spanning (hyper)forests and spanning (hyper)trees. All these results are generalizations of Kirchhoff's matrix-tree theorem. Furthermore, we show that the class of integrals describing unrooted spanning (hyper)forests is induced by a theory with an underlying OSP(1|2) supersymmetry.

50 pages, it uses some latex macros. Accepted for publication on J. Phys. A

References in corpus (7)

• The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
• Simplest random K-satisfiability problem
• Fermionic field theory for trees and forests
• Spanning forests and the q-state Potts model in the limit q \to 0
• Bicoloring Random Hypergraphs
• Renormalization flow for unrooted forests on a triangular lattice
• An exactly solvable random satisfiability problem

Cited by in corpus (18)

• Approximating the partition function of the ferromagnetic Potts model
• Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians
• Multiple and inverse topplings in the Abelian Sandpile Model
• Random spanning forests and hyperbolic symmetry
• Exact solution of the $2d$ dimer model: Corner free energy, correlation functions and combinatorics
• Renormalization flow for unrooted forests on a triangular lattice
• Complete monotonicity for inverse powers of some combinatorially defined polynomials
• Phase transition in the spanning-hyperforest model on complete hypergraphs
• Critical Field Theories with OSp$(1|2M)$ Symmetry
• Spanning Forests on Random Planar Lattices
• Supersymmetric Hyperbolic $\sigma$-models and Decay of Correlations in Two Dimensions
• Hyperforests on the Complete Hypergraph by Grassmann Integral Representation
• Critical Behaviour of Spanning Forests on Random Planar Graphs
• Uniqueness of the infinite tree in low-dimensional random forests
• Multivector Contractions Revisited, Part I
• Dualities between fermionic theories and the Potts model
• Natural Measures on Polyominoes Induced by the Abelian Sandpile Model
• $X$-Ramanujan Graphs