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Fermionic field theory for trees and forests.

Sergio Caracciolo1, Jesper Lykke Jacobsen, Hubert Saleur

  • 1Dipartimento di Fisica, Università degli Studi di Milano and INFN, via Celoria 16, I-20133 Milan, Italy.

Physical Review Letters
|September 28, 2004
PubMed
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This study generalizes Kirchhoff's matrix-tree theorem using non-Gaussian Grassmann integrals. Unrooted spanning forests are represented by a specific Grassmann theory, offering new insights into combinatorial objects and statistical mechanics models.

Area of Science:

  • Combinatorics
  • Quantum Field Theory
  • Statistical Mechanics

Background:

  • Kirchhoff's matrix-tree theorem relates graph spanning trees to matrix determinants.
  • The Potts model is a statistical mechanics model with applications in various fields.
  • Grassmann integrals are essential tools in quantum field theory and advanced statistical mechanics.

Purpose of the Study:

  • To generalize Kirchhoff's matrix-tree theorem.
  • To represent combinatorial objects using non-Gaussian Grassmann integrals.
  • To analyze the connection between spanning forests and quantum field theories.

Main Methods:

  • Development of a generalized Kirchhoff's matrix-tree theorem.
  • Representation of unrooted spanning forests via a specific Grassmann theory.

Related Experiment Videos

  • Mapping to the N-vector model and sigma model for perturbative analysis.

    • Main Results:

    • A novel generalization of Kirchhoff's matrix-tree theorem is proven.
    • Unrooted spanning forests are shown to be representable by a Grassmann theory with Gaussian and four-fermion terms.
    • The fermionic model is equivalent to the N-vector model at N=-1 and the sigma model on the unit supersphere in R(1|2).

    Conclusions:

    • The established framework provides a powerful method for studying combinatorial objects through quantum field theory.
    • The two-dimensional fermionic model exhibits perturbative asymptotic freedom.
    • This work bridges concepts from graph theory, statistical mechanics, and quantum field theory.