Grassmann integral representation for spanning hyperforests

2007/06/30 by Sergio Caracciolo, Alan D. Sokal, Andrea Sportiello · 3 citations
Chemistry · Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Molecular spectroscopy and chirality #cond-mat.stat-mech #hep-lat #math-ph #math.CO #math.MP

paper · pdf · doi:10.1088/1751-8113/40/46/001

published as J.Phys.A40:13799-13835,2007 · 50 pages, it uses some latex macros. Accepted for publication on J. Phys. A

arxiv created 2007/10/16 · openalex publication_date 2007/10/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

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.

Citations

• Bicolouring random hypergraphs
• Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird

Cited by

• Phase transition in the spanning-hyperforest model on complete hypergraphs
• Critical Field Theories with OSp(1|2M) Symmetry
• Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians

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• Grass trees and forests: Enumeration of Grassmannian trees and forests, with applications to the momentum amplituhedron
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